these premises are hard to work with, which is not a complaint about the question exactly. I think that there is allot of interesting things that could come out of this question, but without more back-and-forth discussion it's hard to know where to focus my answers.
I will therefore try to do two things. First, I'll give a simple answer with the presumptions you have written. However, I will then elaborate more on why such an answer is so difficult to give, and why I feel it's partially a lie.
To answer your question, with the premises as stated and assuming otherwise identical to humans, I would imagine males would not be the original target of the process. In fact I doubt 'males' would occur, but hermaphrodites definitely could. I could see the chain of events going something like this:
develop methods to treat birth defects and genetic abnormalities prior to child being conceived. This would happen far faster then it does with humans. Because the mother's DNA is (almost, see below) exactly copied into the child if a mother has a genetic defect the child WILL HAVE the defect, inevitable, except for cases when mutation happens to fix said defect; which is unlikely. Thus, a means to conceive without specific defects the mother has (due to mutation) will be desired
The easiest approach turns out to be to copy DNA from another Pathon, likely a close relative, into the offending mutated area of the DNA of a new 'embryo' in a process similar to IVF & designer babies. Since close relatives would have almost exactly the same DNA (I'm assuming some mutation still occurs each generation, see below, but not enough to invalidate this presumption) they would be safe examples of what the DNA strand should look like. Thus asking mother's and sisters to provide DNA to fix abnormalities you don't want to pass on to your children becomes common.
People decide that they don't want to settle on fixing current abnormalities. They notice that another Pathon has a cool mutation they like and wish they could have that mutation. Thus they start asking less closely related Pathons to provide DNA (or the Pathon equivalent) to get those copies.
Assume that Pathons lack the ability to pull out specific traits (not at all hard to believe, DNA is not that simple). Perhaps the Pathon's version of DNA is even harder to mess with, and you need to copy large portions of DNA from one Pathon to the other. Then copying a Pathon's DNA would require less surgical precision and more of a replacement of large portion of the Mother's DNA. (with the original step 2 the close relatives DNA is so close to the mother's that even a large swap of DNA doesn't mean much, the two Pathons are nearly identical genetically so they are replacing DNA with exact copies of it).
Eventually as this becomes more common the Pathons start to see how positive the genetic diversity provided by such methods are. It becomes common for them to find 'mates' and trade DNA for their children. This starts to become the norm, due to the advantages and ability to lower cultural stigmas (see below).
Eventually you have a defacto hermaphrodite race that usually has young with another mate. Only in this case they may have multiple mates instead of just one. You could then have them force evolution of physical means to make this easier (ie sex, and your 'males'); however, this seems less likely. To reach this point they are good enough at having children via uterine replicator and/or IVF, and would prefer these methods since it's safer for the child and can filter out negative mutations. At this point going 'backwards' to traditional sex seems unneeded, technology does it better. There is a massive expense and difficulty in trying to genetically make a new sex, I can't begin to express how hard this is or how long it would take (good luck getting everyone to keep working towards this goal over hundreds of generations). It's just easier to stick to uterine replicators and doing the DNA exchange via doctors and technology then to change your physical structure.
If you mush have an actual sex you can claim that some eventually focused on providing DNA rather then carrying themselves. Eventually some traits became so common in provider of DNA that they got treated as effective males, and a sort of specialization (sexalization?) occurred. However, it seems unlikely in the presence of modern technology for various reasons. I would need a whole different question with it's own mulch-paragraph answer to go into how to justify going from step 6 to physical change into different species. I do have ideas of how to justify it, but the require allot of justification to make them the least bit plausible.
For this process to work your need a few things.
the ability to genetically transfer DNA from one pathon to another, but not to cherry pick specific DNA subsets. An all-or-nothing approach. This isn't too hard actually, since this species isn't from Earth. Just say their version of DNA encodes differently, and makes it such that you have to transfer huge portions of the DNA to transfer any of it.
The Mother's will not, as a whole, accept having their child contain a fraction of their DNA for no cost, if they did then less of their DNA would transfer to the next generation, and the ones genetically/psychologically inclined to refuse accepting others' DNA would 'win' the genetic race in the end. The easy way to avoid this problem is have the mother's share DNA. If mother A asks for DNA from mother B it will be a quid-pro-Que relationship, where both mothers agree to have a child that will contain some of the other mother's DNA. That way the total amount of their DNA will be be passed on to the next generation, even as they benefit from added genetic diversity. This may result in mother's raising their shared children together
You need to decide how much DNA a 'transfer' between two mother's requires. can a mother accept only 1/10 of DNA? If so why do they accept 1/2 of the DNA, they would be inclined to have their child carry more of their DNA. The easy answer is to work off of point 2 above. Maybe as little as 1/10 of DNA could be transferred at a time, but culturally the norm becomes to have two mother's carry children together and each child carry half the DNA, so that they have shared investment in each child. It may be that other permutations are possible though. Maybe sometimes 3 mother's will get together and agree that each of the three will have a child that contains 1/3 of each mother's DNA etc. The point is I would say that a large, but no where near half, percentage of DNA must be transferred for this trick to work; to justify step 2 & 3 of evolution above occurring, which works better if the child still contains most of your DNA after a transfer), and have the tendency for parents to have a 50/50 split be due to cultural convention and agreement to maximize genetic diversity while still ensuring the same total amount of DNA is spread from each mother to next generation.
Now, lets get into the reasons this answer is not enough to address the premise. There are some issues with the premise itself that I would suggest hammering out first before digging into things like my above answer.
First, despite being a huge fan of evolution and being fully aware of how imperative sex is to facilitate the process, I think it's possible for a sapient species without males to evolve. However, doing so requires extensive world (and species) building to not just justify, but extrapolate from. The species will be different from humans, in many important ways. I can't answer the question as to why males would be desired without first discussing the culture, psychology, and physiology of your species.
To make this species work at all you need some form of added genetic diversity. If it doesn't come from Sex something else must provide it. An actual virus, as you mentioned, wouldn't do this, because there would be no benefit to the virus; it doesn't spread it's own DNA by triggering random mutations in others.
lets consider a somewhat related idea though. presume the parents themselves have evolved some tendency to intentionally trigger mutations in some controlled manner. Since mutations can be the root to long term adaptation and short term survival a form of controlled mutation to help with adaptability may not be harmful. You would risk the possibility of these controlled mutations leading to lots of harmful birth defects though; most mutations are bad after all. However, maybe the species developed a way to control mutations so they only occurred in limited areas of their DNA, combined with some sort of 'DNA verifier" that would quickly exclude bad DNA. Perhaps the mother's get pregnant often, but are prone to early miscarriages as their body realizes this pregnancy has a dangerous mutation and won't survive (human actually do this already, but this species would presumably do it far more often). This can be justified by assuming an enviroment that puts heavy strain on pathogenic species; for instances a huge array of viruses and bacteria which likes to evolve for a specific pathogenic sub-species and attack it viciously; forcing adaptations to be mandatory to survive disease.
This could work, but it brings up a major problem. The Pathons would not be the same. These mutations would change them, but nothing would control the genetic drift; meaning Pathons would become more and more different from each other with each generation. To give an example imagine each generation each child is 1% different from the mother, and that each mother has 2 children.
First generation each child is 2% different from each other. Next generation you have children that are each 2% different from their grandmother, and 4% different from their cousins (almost, it's minutely less technically). By the 10th generation they share 90% of the original parent's genetics, and are only 80% matches to their most distant cousins. by the 30th generation they share only half of their DNA with their cousins. This grows experimentally worse from there. 30 generations is not long on evolutionary scales, but it's enough to have effectively created different species.
You could use approaches to limit this effect, but in the end all the Pathons will be different based off of their parents. Some will be smarter, some stronger, some faster, and some just able to pop out more kids. You will have 50 different 'races' where 'races' actually are different. Now think back to all our racism in the past, and imagine how much worse it would be when there was undeniable proof that the races were different AND a (much stronger) biological imperative to support your own 'race' above all other races. You get racial battles of the most extreme type, and a desire to keep your DNA as close to your mother 'race' as possible which would make one wish to avoid sexual reproduction and getting DNA from a third party.
Put the above idea aside for a second, I'll come back to it, lets address a separate issue. specifically they won't have romance! There is no need for romance, sex, or pare bonding without...sex. Each will have their own child without partners. Their family will be imperative to them, sharing 99% of the same DNA, but that will be a familiar bond, not romantic.
Sex will definitely not be required for pregnancy, as you suggested. Yes there are a few creatures today that reproduce via pathogenesis that still require sex, but that's because they evolved as sexual creatures who only later adapted to be pathogenic, and haven't yet gotten around to removing the need for sex entirely. Without a sexual species existing at all (as you state in your premise above) there is no need to evolve a system where you need a partner to get pregnant.
However, there is an option to address the genetic drift (and racism) issue and lack of romance in one go. Suggest a method in which DNA is swapped between these species, but in a less controlled method then sex. Perhaps regular interaction with another can result in picking up some of their DNA, perhaps even via viruses as you said (though justifying how this is evolutionarily advantageous to the virus takes some work in itself). This would potentially be advantageous as it can help allow genetic mutation to be shared across a community, rather then each mother having to try (and most often fail) at genetic mutations until they found one that 'worked'. It would also limit the genetic drift.
However, this would make the need of the evolution I suggested above a bit harder. It also suggests extreme racial wars between communities that haven't had this 'virus' help them share genetics.
Then again, if you assumed a more controlled and intentional way of sharing genetics without sex you could go from that one to creating sexual creatures with a male...
Meanwhile, we still have the problem that any of the possible species above, other then the last one with it's own evolved intentional genetic exchange method, would most likely evolve males or other intentional means to exchange DNA eventually. It's a powerful evolutionary advantage, someone will stumble upon it.
I'll stop here, because I think I made my point. For me to really suggest a proper evolution we would first need a few species building questions just to hammer out how the species evolved and what it looks like now before I can go into detail about how they would further adapt to creating males.