I have stack-exchanged through my undergrad math program. Am I likely to succeed in mathematics PhD programs?
Yes! In fact, I think you're well on your way to doing better than your peers! Taking longer to understand something isn't something to be proud of! There's no need to reinvent the wheel. If someone can help you understand something, you would be well-advised to make use of them. In the same way, you would be well-advised to attend the lectures, thereby getting help from the professor, instead of staying at home with the ZFC axioms and attempting to derive the whole of mathematics from scratch.

My experience asking questions on Stack Exchange also indicates that simply writing the problem in a form which others can understand is a great help in clarifying my own understanding of the problem. Indeed, I've solved some of my problems simply by beginning to write a (never asked) question.

Of course, this doesn't mean you should outsource your understanding to others. You should make it so that you can explain the concept to next year's students without help (answer some other SE questions while you're at it!). But even then you should feel free to ask for help with next year's problems.