You seem to know a fair bit, maybe you have a math PhD yourself?
Nope. Tried and failed. Lack of any real ability was one factor; others were family, lack of funds, lack of enough background to attempt a research problem, and an incompetent advisor. Hardly unique. I've known a Fields Medallist who almost didn't complete his doctorate because he was making no headway; he thought of becoming a C programmer.
One should think twice before attempting a math Ph.D. Universities are hardly hiring anyone, and when a prof retires, replace him with one or more adjuncts. The competition for jobs is fiercer than ever. And one should always keep in mind what Courant said sixty years ago: " In the future, the only mathematics that will survive wil be related to physics, biology, and economics."
If one does decide to go for the Ph.D., one should make sure to have taken a whole swathe of first-,second-, and third-year grad level courses at a major university so that one can attempt a thesis in a mainstream area of mathematics (e.g. modular forms or cohomology of groups or algebraic number theory); if one's dissertation is in some peripheral backwater area, with a dissertation title like "Virtual representations of ghost pseudo-loops," it'll be even more difficult to get an academic job, and the area itself may be a dead end. Having completed these courses (and of course having passed the qualifying exams and got through the other bureaucratic requirements), choose an advisor with care. The advisor shouldn't be some old fogey, well past his prime, who has no idea what's actually going on in the world of ideas. Nor should be some never-has-been, who, except for a handful of obscure papers, has nothing to his credit. He should ideally be a mathematician in his prime (say between 30 and 45), and more than just a narrow specialist: he should have a bird's eye view of neighboring disciplines, and have a feel for the way the whole area is developing. He should already have supervised a thesis or two so he has some idea of what a research student is capable of. He should also be able to encourage students (there are first-rank mathematicians who destroy grad students).
The dissertation itself defines the research student in the eyes of the mathematical community. Of the 20-50% of grad students who complete their Ph.D., 90-95% never do any further research. Of the minuscule percentage who do, their subsequent papers are usually related to their doctoral work. So make sure the Ph.D. is in a mainstream area.
Most dissertations are unremarkable. The exceptions would be Serre's thesis, which introduced spectral sequences (thus revolutionising algebraic topology, and earning him a Fields Medal), and Donaldson's thesis (on 4-manifolds, which revolutionised the field (until Seiberg-Witten theory came along) and earnt him not only a Fields Medal but a chair at Oxford at the age of 26).