I have no idea what it's like in the US, but in my part of the world doctors aren't salesmen. You want to see a different doctor? Fine, that's your choice. You're going to be stuck in the cue for another couple of hours but sure, go for it.
For most of these entry to mid level client facing roles why would you hire someone with tattoos/piercings/anything to make a client potential not choose you when you can hire someone without any of that? It's not like their is a shortage of talent or applications for positions in stuff like law, investment banking, trading or consultancy. I think Mckinzie gets something like 3x as many applicants just from university of Pennsylvania as it needs for positions world wide every year. They don't want to risk loosing a client because someone didn't like their consultants tattoos. For IB interviews in the UK they usually send half of the group home at lunch and then continue interviewing the other half. They always sent home the people who couldn't be fucked to shave or dress nicely.