I think we overvalue the uniqueness of the individual in this hero worship of the lonely genius, as if a flower is independent of the earth from which it grows.
But then again, it's true that there are exceptional stars, singular phenomena that cannot be explained by the sum of its parts. I suppose that leap, the surprising distance between what was given and what the individual made of it, is what we call genius, talent, luck or hard work.
We can see common evidence of this in how every person we know differs in their general abilities across different problem areas.
But we are so used to that we are probably under aware of how differently we may all think.
Potential geniuses would be people whose internal abstractions are uncommon and happen to fit important under-solved problem areas.
If their uncommon insights are paired with healthy brain biology, a drive to identify and solve interesting problems, and some luck, we get a genius.
It's more obvious in athletics, where there is clearly natural talent, but the equalizer is the support network of parents funding practice, driving to games, getting into competitive leagues, etc. I don't see why the same wouldn't apply to intellectual talents as well, even if the differences in natural talent are less obvious than in athletics.
1. effort, usually from youth, that nobody knows about so it appears to be innate
2. the effort is motivated and guided by some mentor(s), usually people with serious qualifications, like your Fields medalist uncle deciding to take you under his wing, after you said "math is fun :D" one time at 7 years old when he told you his job was "to do math :)" upon you asking him as children like to do, and turn you into a Fields-winning adult
This reminds me of those people who pretend they're a genius because they can guess the day of the week if you give them a date, when the reality is that anybody can learn to do that because it's just an algorithm[1] that you can calculate in your head and practice to the point that you come off as if you have a photographic memory or something.
[1] https://en.wikipedia.org/wiki/Doomsday_rule