Not necessarily. Remembering all of the multiple representations of the beta function, for example, is probably aided through the use of flash cards. You can still use such representations without necessarily having to go through and derive them from scratch, whilst still understanding what the beta function is. Ditto for the many trig identities.

Similarly, there are often underlying assumptions that can be tricky to remember in the moment, e.g. certain log laws only holding for the absolute value of the argument. There's a combination of both understanding a tool to begin with, and remembering various equivalences, representations, and underlying assumptions that makes math difficult.

Part of memorizing proofs is also just increasing your exposure to certain ideas, because maths is one of those subjects where there's no real substitute for time spent thinking about something (i.e. mathematical maturity).