No it wouldn't. Most math PhDs require six or so courses before the research - you can begin with analysis or abstract algebra for proofs. The current system does have unnecessary redundancy.

> Like programming, writing proofs is a skill that takes practice to get good at. If you were to suggest that we shouldn't teach programming until graduate-level CS courses, you would be laughed out of the room.

Programming is more comparable to the computational mathematics that I've already mentioned should be in undergrad; more to the point, a lot of computer science theory need not be taught in undergrad either. Computer science theory is much closer to proof-base mathematics.

> I think the best mathematicians in history started learning how to prove things as children.

Approximately no one is one of the best mathematicians in history. That's an attractive idea, but I think it's more realistic to assume they were extraordinarily precocious than that proofs are easier to digest at a younger age. I could point to people who mastered computational mathematics at a young age too (like Einstein, at 14), but that doesn't say anything about which is easier to learn. I'm talking about which is more widely applicable for practical work.