Let's start with an easy one: https://en.wikipedia.org/wiki/Axiom_of_choice

Lots of drama around the ABC conjecture in recent years

Well for medicine, there's alternative medicines like chiropractic and homeopathy. According to some people, these are snake oil and quackery, but many others swear by them, and many healthcare systems even provide these treatments despite the complete lack of scientific backing.

Significant (as in, directly impacting millions of people) geography-related disputes are in the news literally every day.

Medicine - COVID vaccines.

Science - global warming.

Unfortunately, humans and their politics are intertwined into more aspects of life than one might deem rational.