I don't see why if you are independently wealthy and don't need to make a living and have as much free time as you want, that you wouldn't be able to self-study.
I have a PhD (granted, in topology, not string theory, though my work was very close to that of some string theorists, almost to the point of being scooped by them), and I'm not sure supervision was all that vital, at least in attaining the level of knowledge that I have. My adviser did provide some guidance that allowed me to get unstuck at a few points in my thesis (some difficult little points for a newb like me to get through on my own), as well as help with learning to write, but that's just a question of producing the thesis/research/publications, rather than simply acquiring knowledge (and for the writing part, at least, there are other ways to learn that). In some ways, it was an advantage, but there's also this whole "drinking from a fire hose" phenomenon in grad school that I'm not sure is the optimal way to learn. Many of my friends from grad school would agree that just reading a book would have been better than some of those classes. I think being free from the pressure of grad school would have had considerable advantages, although there would have been a trade off. Given that I failed as a mathematician anyway (though, perhaps more on the teaching front), I think it would have been less traumatic if I had just studied on my own, the only issue being that I would have had to make a living, which would not have allowed time for it (although I did have to work part time for the department, anyway). I would have been much less overwhelmed and would have been able to have much greater retention and understanding of the material, due to not having to drink from the graduate school fire hose. The trade off would be that I wouldn't have been challenged as much and therefore forced to overcome some of my weaknesses. And I would not have been as well exposed to what was going on in the community (given that I was a failure, anyway, not much advantage, there). There are also subtle things that you do get from some of those graduate school moments that you wouldn't get by yourself. However, given the high stress environment, I'm not entirely convinced I wouldn't have been better off on my own. The stress just about killed me, and I wouldn't have done that to myself. And I don't really think it was necessary, for the most part. Maybe some is good, but you can try to simulate that and give yourself deadlines and stuff, if you need to. You can try to make a sort of unrealistic goal of reading some book in a certain amount of time, just to see if there are ways in which you might be able to make your learning more efficient and think about whether you might be spending too much time on certain points when it would be better to move on (that was one of my issues that grad school helped with a bit).
This is the perspective of someone who had a very hands-off adviser, I suppose, and maybe that's one reason why I didn't do that well (perhaps, my bigger failure was due to teaching, though--they said my thesis was good, actually). Maybe I would have done better with more guidance.