SDOH are the conditions in which people are born, grow, work, live, and age.
www.cdc.gov
LGBTQ+ identity is a pretty important part of social determinants of health... which absolutely affect how patients do. So the argument that it's not "healthcare related" doesn't really work.
I can't really see any distinction important to your argument between a cross necklace implying "I am a Christian" and a pride pin implying "I support the health and personhood of LGBTQ+ individuals".
Additionally, in practice, you'll find that if someone asks you
literally anything about the pride pin you're wearing (
out of ignorance, fear, disdain, hatred, literally whatever), you can diffuse the situation by saying something like "I wear this pin because I want to make sure my LGBTQ patients feel comfortable so that I can properly treat and care for them" and you're done. If it offends a patient so much that it significantly impedes treatment, you can take it off when seeing that patient, but the likelihood of that happening is pretty much 0. No patient or family member is going to draw you into an argument about it. If an attending or admin is going to have a problem with it, it would have to be with wearing pins at all and not just the pride pin because otherwise that would be a huge potential HR headache for them.
Bottom line:
this thread is so incredibly hyperbolic even for SDN