But when we are talking about unsolved problems, we are in the land of vanishing small probabilities in any case. Could it not be that the very fact that someone has been trained in conventional mathematics a hindrance to them resolving an unsolved question, and someone thinking in isolation has an advantage?

Here's Grothendieck, for example:

“In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”

I can certainly see this happening: someone away from the pressures of publishing, the drudgery of admin tasks, the frustration of applying for grants or department politics, or the fear of looking "bad" to their peers etc. devoting their time to mathematics out of pure joy, play and drive and coming up with novel definitions and assumptions that those in establishment mathematics, out of pure sociological and psychological reasons, have not even dared venture into.