What's the underlying test?
Is atheism something that even can influence such? Atheism is the lack of belief in any gods. That's not even a lack of belief in the supernatural, to be clear. It's certainly not a belief that one should do anything.
If I started a thread here called "What would be the positives if we had more atheist politicians?", I'm sure we'd get lots of responses approving the idea that the influence of having more atheistic politicians would probably have indirect benefits.
Sure! To be clear, though, that's not because atheism is pushing or supporting actions. In fact, it's much the opposite. It's that those politicians assumedly wouldn't be pushing religious agendas that are frequently not based in fact or the best interests of all (in fact, a number of the current overtly religious agendas being pushed in the US are rather distinctly net negatives when it comes to their entirely predictable effects). Much of this sentiment, such as it is, is backlash to politicians doing exactly that, though, not because atheist politicians are inherently better. Going beyond that, being meaningfully represented in the first place is usually a distinctly positive thing, even if a disproportionate representation is generally not as good a thing.

And?
Yes [we do need more atheist politicians] because they are more likely to be educated. There is an obvious correlation between atheism and intelligence levels. Therefore increasing the chances of a productive president if he/she were an atheist.
If we were dealing solely with averages, this might hold up. When it comes to politics, we aren't so much. That goes into a notably tangential discussion, though.
Atheists are also usually more moral. I know, it's a shocking thought but think about it. Whenever you see an atheist arguing over a moral situation like war or homosexual rights they're usually arguing for equality. Atheists want equality for everyone because none of us have a reason to hate anyone, whether it be sexual preference, ethnicity, nationality, or whatever else Christians hate people for.
This answer, too, is a highly questionable assertion. Besides what was alluded to about averages and politics, there are plenty of theists and religious people who do much the same. It's certainly true that they're less likely to show up in the reaction-seeking media business models, though.
If I started a thread called "What would be the negatives if we had more atheist politicians?", I'm equally sure that atheists would respond as they have here: "What, atheism? It just means lack of a belief in God or gods. How can there be any influence?"

Half true. But that was already gone into in more depth above and elsewhere in things that you seem to want to ignore.
It sounds a little naive to me, a "no TRUE atheist" fallacy.
You've been sticking your fingers in your ears and screaming "LALALALALALA I'm not listening!" by the look of it, then. There are lots of "bad" atheists. Much like there's lots of "bad" theists. They're not good or bad because of theism or atheism, though. Their more comprehensive philosophy, religion, or world views are far more pertinent to such things. With that said, it might be beneficial for you to actually understand what a "
no true Scotsman" fallacy actually means, rather than continuing to embarrass yourself by continuing to repeat your distinct misuse of it when no one here at all even suggested that those in question weren't "real" atheists.
Like it or not, I think a case can be made for modern atheism coming with lots of existential baggage that does have real world political implications, for good or for evil, even if they are indirect.
On the other hand, "Garbage In, Garbage Out" still applies. We could certainly have a discussion about the circumstances of "modern" atheism, to be clear, and I've been rather doing so on this end, at least, much as I've been limited to some of the most basic concepts because of your clear failures to accept that atheism is not, for example, in any way supportive of totalitarian governments. Rather, it's exactly because atheism
cannot push some opposing set of values, authority, and beliefs that some totalitarian governments push it to increase how well they can control their population. That still leaves what's actually being pushed as the important thing, not the thing that cannot push anything, which makes it rather dishonest when one keeps trying to focus on that which cannot push anything as if it was as important or more important than that which actually is being pushed.
Ask Falun Gong practitioners, before their organs are harvested.
Ooo! Ooo! I can play at this too! Ask the Jews, before they were brutally murdered en masse in the name of Christianity! Ask the Rohinga, before they were brutally murdered in the name of Buddhism!
What are my fundamentally wrong premises? Please quote me, if possible.

Plenty have been touched on in more directly relevant spots. Your attempts to treat atheism as on par with a religion or a comprehensive world view, though, are some of the most basic. Atheism would be on par with the very general use of theism. The Chinese's version of State Communism would be more on par with a religion, on the other hand.