It helps a lot, but definitely isn't a prerequisite. Having some competition math experience myself has helped me progress quite quickly in CP, as lots of the problem-solving intuition is similar. However, even doing advanced competition math isn't a free pass to any CF title; only hard work and knowledge of algorithms will help you then.

If your end goal is CP, then just do CP. If you want to do math contests too, go ahead and do both. If you are dead set on optimizing your strategy to be the best CP programmer that you can be, then you should do what you want to get better at. But I don't think there is any great reason to do either besides for fun.

Learning math and proofs is very helpful for competitive programming but you don't necessarily need the skills from competitive math. I have very poor competitive math skill, but I learned what I know through math courses, books, and exercises from books. There isn't one correct path, and you should do what you enjoy.