As I said in one of the threads, you're talking about relatively small numbers — four or five labeled "Martians" because they intersected at a particular point in history with particularly overlapping skills that were useful for the United States during World War II. There were no doubt other Hungarian emigres who would be worth talking about if we were truly talking about Hungarian emigration and its impact, but we're not, we're talking about these particular people (Szilard, Wigner, Teller, Von Neumann, and sometimes Von Karman) who happened to be friends and also who happened to have a big impact on US weapons work. We could just as easily call the Polanyi brothers (Michael and Karl) Martians if we wanted to, but their work didn't overlap with those five so we don't. So there is a small-number selection bias here that is going to resist any kind of meaningful "historical explanation" beyond very vague ones — e.g., they weren't actually emigrating immediately from Hungary, but from Germany as a professional way-point. Why were they in Germany? Because that was where you studied if you wanted to excel in mathematical physics, which is what all four/five of the "Martians" did, in essence.

We could just as easily say, "wow, there were several really important Polish scientists on the Manhattan Project as well!" (Ulam and Rotblat come to mind, I am sure there are others.) But we don't. Why not? That's a different question — why the Martians got talked about as a special little group then and historically — but indicates how arbitrary it is. (Part of the answer to why they got focused on is, I think, both the total intelligibility of their language to everyone non-Hungarian, but also that a few of them, notably Szilard and Teller, had HUGE personalities, and one of them, Von Neumann, was so smart as to be a genius's genius. None of which will be explained by appealing to historical factors — these are individual factors.)

In other words, I don't think the answer is to "why did these four/five people excel in the United States during WWII" will be found in appealing to the superiority of Budapest public schools. (They didn't go to the same schools as one another, anyway.)

As for what role their Jewishness plays, it's partially the fact that Jewish scientists were (because of the Nazi threat to Jews) uniquely motivated to emigrate and work on Allied weapons projects (and you see this bias across the board), but also that, then as now, Ashkenazim culture highly values academic education when it finds people who are talented at it, and pushes very hard in that direction; at that point in time, that could mean mathematics and that often segued into physics. There is no magic to that — other ethnic groups do the same thing and achieve similarly disproportionate results in many fields of study if they are not being systematically excluded from them (and the Jews were systematically excluded from some branches of science; the places where they thrived were places they were not — theoretical physics in particular was considered to be less prestigious in Germany, which is why you see so many Jewish contributors there and not in the more-prestigious experimental physics during that time period).

But in any event, you are treating "Martian" like it is a formal analytical category and not an in-joke between people who knew and liked each other, and trying to subject it to a sort of rigorous historical accounting. That will not really get you very far, any more than trying to push a similar line of analysis to the membership of the "Rat Pack" in the 1960s will get you ("if Sammy Davis Jr.'s Black Jewishness was a factor, why weren't there more Black Jew Rat Pack members?", just to show the absurdity of this).