The cult believes there are already enough souls existing in the world in order for genocide to trigger the event. There may be always enough people existing, it's just a matter of how many can be killed in a relatively short time frame. Never in the history of humanity could that point be achieved due the lack of technology and the inability of radical ideologies and beliefs to spread as needed. Throughout history the cult may have always tried its best, but it was never enough (the goal is never obvious - after all it's a belief).
How their belief of souls could work? Their god split up a significant portion of his power (soul) in order to create humanity. Well, that caused something like a gaping wound to him - or rather an eternal balance mechanism which slowly seeks equilibrium between the total soul power of that god and humanity. You could say god kept 50%, humanity has 50%. Whenever humans die, he regains some power. But that is only temporary - and the only way to regain sufficient power is a relatively short time frame of sufficient mass extinction. He just has to reach, let's say, 80% once. The 20th century as a whole was a great attempt - fascism and communism - tasty mass murders, world wars, genocides left and right, but it was just not enough.
The reason why that god is "powerless" in that state of equilibrium is because his power is tested against that of the plane of existence. Well, he imbued humanity with half his power which translates to souls for humans, and humanity is bound to Earth in the plane of existence... so all his influence and powers to affect anything in the entire universe are blocked by humanity existing and wielding his powers (having souls).
With the mechanism of equilibrium the absolute power/souls can increase and decrease at will. Humans can generate more souls through reproduction, and that god can have increases and decreases as well. But that does not matter - however it allows for scalability and renders it timeless.
As long as the equilibrium exists, he can basically do little at best. No fancy god powers, just whispering some people some things, maybe a bit more. He may traverse through time and space, but the creation of humanity bound him and his powers to Earth eternally... unless the cult can restore him.
The mentality of the cult:
Not knowing what magnitude of human extinction is necessary to restore their god and to receive the salvation they seek for millennia, throughout their existence they sought to establish world wide events of wars and destruction. Due to their low numbers - it's not easy to gain followers after all - they could never exert sufficient influence and were forced to act in the shadows.
They are aware of the power of technology and the potential for destruction it bears. They were highly impressed by the deeds of the 20th century and learned a lot from history - in their own perspective. They align with every ideology which has adherence to high state power, control, tyranny, totalitarianism, dictatorship, collectivism and consider them ideal vehicles to pursue their goals of rapid human mass extinction. The core concept of state power and ideology was long known to them, but with the 20th century they saw their goal to be in close reach like never before. But at the same time, some of the cult members consider it to be a time of urgency, as capitalism, individualism and freedom have lifted tremendous suffering from those benefiting from it, lifting living conditions within mere 1-2 centuries like never before in human history. That has to stop!
Their set of morality does not comprise of being truthful, valuing life, valuing individualism, or freedom. They despise life and existence itself - human existence in itself is a sin, a betrayal of their god - yet they are willing to make the sacrifice of living and acquiring wealth and/or power in order to pursue their goal. Every tool which serves their goal is justified - the goals justify the means is not just a justification for amorality to them - it's the paradigm to live and act by. Thus "sacrifice" is a core element of their religion - to them their existence is a sacrifice, a necessity, but also a torture of their god, who in turn himself has to bear the sacrifice of enduring the pain even from his followers. Another sacrifice for cult members is having to lie to everybody and to hide one's true self forever, and to hide the pain which is existence. It has a ritualistic meaning to them, especially among their kin, thus also blood and shedding blood, which is also a metaphor to death.