And worse, by spending time on solved problems and refuted ideas in public, you can easily destroy your credibility with those that could help you.

This is a serious issue with how people like us, that have interdisciplinary interests, interact with and are respected by experts in fields touching on our own. Those that study, for instance, epistemology, view those that study, say, probability theory, fairly negatively, because they keep hearing uninformed and stupid opinions about things they know more about. This is especially bad because it happens instead of gaining from the knowledge of those experts, who are in a great position to help with thorny issues.

True. But there are also personal benefits to working on problems (increased cognitive ability, familiarity with useful methods, etc.) that arise even if the problem itself is already 'solved.'