How would a religion centered around summoning a god of destrucion to destroy the world attract new members?
What incentives or arguments would convince a mostly sane population to go along with their plans?
There are lots of religions and cults that already involve the world being ended by the central deity. Most familiarly to many readers, Christianity. It's a core aspect of Christian belief that God will end the world at the Day of Judgement. (Yes, Satan might get the blame but God's omnipotent, right, so if he didn't want the world destroyed he could just prevent it.) So, using Christianity (and some other religions) to provide some examples, let's look at some of the reasons people may be attracted to such a religion:
Belief in belief. (This isn't a tenet of a religion but a description of a phenomenon of the human mind.) People believe they believe things that they actually don't believe at all. They don't notice the difference between belief in something (the unshakeable conviction that something is true regardless of lack of proof or counter-evidence) and thinking that it's good to believe something. As an example, this explains religious types who fervently preach against sin and the eternal damnation that, in the absence of repentance, will inevitably follow, while themselves continuing to commit all manner of sins. Either they're looking forward to a nice long brimstoney bath or they don't actually believe what they think they believe. (Equally it explains Buddhist monks becoming involved with nationalist violence in Thailand, which runs counter to the message of the Buddha as expressed in the Noble Eightfold Path.)
Personality. There are plenty of apocalypse cults based on the hypnotic personality of the founder and a few brainwashing techniques: Aum Shinrikyo, Branch Davidian etc. Whether these scale well beyond a number that the founder can have direct control over is a good question: although not really an apocalypse-focused cult, Scientology has managed to develop a structure that extends an originally personality-centred organisation to a significant size.
(Follow-up: please see the discussion for some examples of how people might go about defending a religion involving the eventual destruction of the world. There were some reasonable criticisms of the first version of my answer, which I have since tightened up, but there were also quite a lot of replies that did include the kind of arguments that I have described above, most notably attempting to deflect attention from the awkward bit about the end of the world and focus on the nice bits about creation and love.)
1: Promises of an afterlife during times of crisis.
Ever seen the movie The Mist based on the works of Stephen King? The people in it are desperate and feel out of control, eventually they start following a christian based doomsday religion in exchange for some hope of an afterlife.If those gods of yours would give them a nice place in heaven it might be worth it to them.
2: To build a better world in it's ashes.
If the destruction is not meant to be permanent people could be inclined to help destroy it to start the re-birthing process sooner. Take our global warming situation... won't be that surprised if some would start preaching genocide to reduce the world population to make it more habitable. (What the bad guys in the newest Godzilla movie wanted to do)
3: What we do now, we push the problems ahead of us.
Most people barely care what happens after they are dead so why not go for it if it has other benefits like increased lifespan or riches.
4: Hiding from the actual crazies.
Most of the world is already insane, so if a doomsday cult becomes dominant and already starts killing everybody who disagrees it might be better to just join them to at least survive until the actually apocalypse begins.
Quite some founding books of real world religions are, to put it mildly, pretty harsh with their prescription toward those who not follow the book while promising a good reward to the observers.
Those unfaithful persons are said to be facing the wrath of the deity described in the book, either in the afterlife by different kind of punishment, on in the present life by authorizing or prescribing the observers of the faith to fight and discriminate the non believers, literally destroying their world.
Not surprisingly those religions have also been a strong uniforming factor in the regions where they were majoritarian, "convincing" many to convert for the pragmatic sake of a quite living.
Start with a cult like the Thuggee. Kali is a complicated goddess and one of the beliefs was that a minimum level of death needed to be maintained or she'd killed everyone. Then you get a judgemental breakaway group who get upset their religion is being suppressed so they decide she needs to be summoned instead of pacified.
A mainstream religion with apocalyptic prophecies that lead to a glorious new world where believers will be saved, but first you have to have the end times. This is the ISIS model where they attempt to accelerate that final battle.
The general theme is that the believers are going to be saved from whatever happens.
(I've gone into a lot more detail in the past on destructive gods.)
The general practice used throughout history is to make the following argument:
This is actually an easy distortion to make within any proselytizing faith. Proselytizing faiths — faiths that actively try to convert new members — already share the first two principles of corruption and Divine anger, but believe that the solution is to bring the corrupt into the faith where they can experience the loving side of the Divine. Proselytizing faiths are hopeful and compassionate (if frequently quite annoying).
However, all it takes to convert a proselytizing faith into an apocalyptic faith is a leader or set of leaders who ratchet up negative emotions: fear, hatred, jealousy, bigotry, etc. Focus on the corruptions of the outside world; graphically depict the horrors that the Divine will visit on the wicked; cut the congregation off from outside influences by painting all outsiders as contemptible, condemned monsters. Then it is easy to convince the congregation to take up violence, first for their own self-defense, and later for the glory of the divine.
This is a pattern that permeates Abrahamic faiths, and has correlates in Eastern religions and the secular world. It's easy enough to imagine in almost any context.
All of the major Western monotheistic religions build on more or less on the setup you describe. They just use better marketing terms.
Rather than "god of destruction", they say "all-powerful god" (whose profile includes destruction on top of everything else).
Rather than "destroy the world" they say "final judgment at the end times" (which involves removing everyone from the world anyway).
Include a doctrine which paints other religions as being evil (i.e. first commandment). This helps to convince your followers that you are the only good guy. And they get into trouble for asking questions to the contrary.
Of course, they don't require followers to assist with "summoning", but adding this in is as easy as including explicit summoning instructions for your followers to carry out.
These religions are also very good at getting converts. How? First, by actively seeking out converts (Christianity, Islam). Second, by promising its followers big rewards which, coincidentally, the religion need never account for, since delivery of payment is deferred to the afterlife.
IMO the most obvious way to sell destruction as a good thing is by claiming that whatever comes after it is better than we have now.
Consider the song Destroy The World Or How To Combine Love And Misanthropy, by King Satan. It starts like this:
Long time ago I had faith in mankind
Long before I witnessed the travesty of it
We are what's wrong with the world
Ever so rotten and full of dirt
Destroy the world, destroy everything
Destroy the heavens and make this end now
Destroy the world, destroy everything
Destroy the heavens, leave this planet
To make the world a better place
Is to dream of the dawn of a nuclear war
Push the button of human extinction
And take us all closer to "god"
I think the sentiment in those lyrics is quite clear. Let's end what we have now so that something better may arise.
To minimize suffering
The world is nothing but suffering and hate. There is an afterlife, but it's nothing but misery and torture. There is no "good" destination akin to Heaven.
This world will end, whether by this god or the natural death of the universe. But the longer it goes on, the more damned are born and the more souls will be subject to eternal torment. Better then, for the sake of the human race, to thumb our nose at the gods and natural order of things and to allow the god of destruction to end it all.
Maybe when it comes time for the gods to etch out the next universe, they'll remember our rebellion.
Religion is, by it's very nature, transcendent. The world and the concerns thereof are not the concerns of religion. Religion answers the quest for the highest good by pointing beyond the world of our physical experience. Therefore members of a religion may very naturally anticipate the end of this life (or the world) with hope.
There are interesting theological questions around how the world itself is viewed, and each religion answers such questions differently. For instance, a gnostic Christian might reject the material world as evil, while an orthodox Christian would uphold the inherent goodness of the world which is passing away. It sounds fun to flesh out such details (don't forget to include internal conflicts within the religion itself).
Obviously contemporary American religious thought won't yield much good source material, because modern American religion is almost always hedonism in disguise; but it would be interesting to look into the Tao Te Ching, some writings of the ancient Christian ascetics, and some literature about Hindu gods of destruction for a wide range of perspectives.
The basis of a positive arguments used to attract followers would be that the coming world is better than this one. This is an easy argument to make given that in our world: 1) people tend to think the world is bad and 2) they also always seem convinced that it is getting worse, in spite of many efforts to make it better. It therefore shouldn't be hard to draw the conclusion that the world must be remade. Of course, a nihilistic argument could be made that life itself is bad (see any bad guy in any movie ever).
A pseudo-Buddhist argument might go something like desire causes us to suffer; the pleasure does not outweigh the evil of the suffering; it seems impossible to not desire while in the body; therefore all desire must be extinguished by the apocalypse after which we will …
I guess, in general, I think it is harder to argue for the goodness of the world than it's badness. Therefore I don't think it should be hard for you to construct convincing arguments for it's demise, be they hopeful for a better existence or not.
Wait, are you assuming people are sane?! Seriously, that one takes the cake!
What kind of assumption is that? Have you studied sociology?! How about Politics?! Geopolitics?! History even?!
That aside, there is a certain freedom to any sort of apocalypse cult, whether causing the apocalypse or just believing one is coming. Imagine not having to worry about how you are seriously screwing future generations, because to put it simply, there are none.
Additionally, one would presume this death deity will come whether they worship him/her/it or not, so if they worship said deity, they will get the benefit of not having to worry about consequences, and receive favorable treatment by not being tortured to death? That's the deal of a lifetime!
This has a vague similarity to the current and past climate change beliefs. There's a lot that goes into it, including money, like Fels mentioned in a comment on the original Question.
Once known as "global warming" and other phrases, climate change is a contested topic. I could go into a lot of detail about how it's essentially a proven fact, but that's not what this Question is about. It's asking about the opposite, so I'll concentrate on the deniers, for a while at least.
Some people deny that climate change exists or accept that climate change is happening, but is not caused by humans. Because there's such an overwhelming amount of experts and data that say climate change exists and that it is caused by humans, denying it has almost become a religion in it's own right in the real world today.
I've talked to people who fervently claim that nothing has changed, weather wise, in their life or even in the past 100 years. Or they claim that what has changed is simply a different pattern than we've recorded since we started keeping detailed records. I'll split this in two parts here, so we can see the differences in thought pattern.
The complete deniers say that nothing is changing and will argue that it was no worse than when they were kids or when their grandparents were kids. They bring up vague memories or stories about hot summers and cold winters, yet have no real data behind it. They more make assumptions based on current affairs than anything. When shown actual historical data, they will point to the worst bits and say "those were the times", regardless if it was or not, as well as disregarding the frequency of the extreme temperatures compared to today. This is a complete belief their personal experience outweighs empirical or statistical data.
These are also the same people who will search for other people that have the same experience to prove they are right. They will even mention the approximately 30k scientists that signed a petition saying climate change doesn't exist. It doesn't matter if you have a fact checker site say the petition is fake (because of a variety of sketchy details about the petition), a political site say it's fake (because of the political maneuverings of people signing it), or the reliable site that reports the how real scientists made their observations and made professional decisions about it. Because the deniers won't believe anything but their personal experience and often refuse to even consider anything against their experience, it has the weight of a religious belief.
What do they get out of believing nothing's wrong? Hard to tell, but it could just be they are hard headed and simply refuse to believe they could ever be wrong. It could be they just believe only what some people/leaders tell them, and no one else. This goes into the rabbit hole of what do those people get out of it. Some of them are business-people in industries that lose money by having to abide by the rules and regulations of government agencies like the EPA. They have to spend money to clean up their factories, the mining of minerals, and much more. They may even have to pay to clean up spills or previous harm they did to the environment before the laws changed. "Since there's nothing wrong, we don't have to spend billions of dollars to fix what's not broke."
These are people who simply think the current changes in climate are out of human control and there's nothing we can do about it. They ignore the fact that, due to EPA and other regulations, smog isn't as much of a problem in certain cities anymore, rivers rarely catch on fire anymore, and even the ozone layer is recovering, among other reasons we've made changes to laws to clean up pollution.
They cite the same exceptional years as the total deniers as proof the weather isn't certain of anything, even in the face of records showing the hottest years on record have almost entirely been the recent past and that the trend is continuing to go up.
This isn't as ardent of a religion, but because their minds can't be changed by facts or arguments, it's still a deeply embedded belief and unlikely to change.
Again, what do people get out of this? They get the reassurance they aren't the problem and that worrying about it won't do them any good. They are also assured that the natural progression is for the weather to go back to what it used to be and that everything's fine. This group of people also get their information from the same people as the total deniers, except they aren't as ardent of believers so they get a slightly different message because of it. "Since we can't do anything about it, we don't have to spend billions of dollars to fix what's broke."
This covers a wide range of people. There are those who simply accept that climate change exists and that humans caused it and can fix it, but aren't doing much if anything to fix it. There are also those who are doing what they can to fix the problem, activists who do a lot to fix it and encourage others to fix the problem, people behind the scenes cleaning things up fairly silently, as well as those fervently pushing to get things changed to prevent the end of the world, to include getting more laws to prevent pollution.
This can be another religious belief system, but to the effect of wanting to prevent the end of the world. From rallies to compost heaps, to speeches on the global forum to town halls, massive marches in capitol cities to setting up recycling centers, there's a wide range of things this group does to evangelize their cause. They also recognize, research, and fight against conspiracy theories surrounding climate change.
Because climate has seemingly such a polarizing effect on people, believers have had to take on essentially a religious stance in furthering their cause. They get criticized anytime they make any sort of mistake or miscommunication, so many feel the need to be absolutely perfect in their defending their beliefs. They also have had to learn to be just as loud and unbending in their devotion as the deniers. Because of this outward display of their beliefs, it can be considered a religion by observers. "If it walks like a duck and quacks like a duck, it's a duck."
We can ask the same question about this group as the other two groups. What do they get out of it? Well, that's also a hard one to answer. Some believe they are doing it for their own good and health. Some believe they are doing it for their kids, and maybe even other's kids and future generations. Some people believe they do it to save flora, fauna, and for other less tangible reasons.
Having a religion that wants to hasten the end of the world doesn't need a logical reason. Latching onto anything that scares people and telling them your religion solves that problem is enough to get a large amount of people to believe. Getting confirmation bias from their own experience goes a long way to endearing them to your religion. You don't even have to get to a tipping point* with this religion to get mass appeal, you just need a large enough group that it appeals to any part of the population. With a planet as populated as Earth, having 100k members in a religion isn't even 1% of 1% of the total population, yet it's still a fairly significant amount of people, due to it still being the same number as some medium sized cities.
*A tipping point generally means that there's a point where all of a sudden there's a massive influx of believers, which you don't specify as a requirement, just that the religion exists.