In another thread (and I think you're only really on this CDC thread) people are talking about how their hospitals are getting the vaccine but are limiting it to only current patients.This is one of the things about this that sounds like it doesn't make sense but actually does- why would places like CVS, and Kroeger, and Walgreen's be better at this than The Mayo Clinic, Johns Hopkins, and Cleveland Clinic?
Simple answer: it's what they do.
When you look at it the same problem exists. Both ways (pharmacies and hospitals) are competent in administering the vaccine although I believe hospitals are having to train people who have never done it before still. But if you're limiting the vaccine to only current patients and Walgreens and CVS are limiting to long-term care and I think the other poster mentioned something about not being able to share (and I can only guess that means not holding a lot of vaccine back in one location) it's the same problem just swapping who it is.
I don't disagree with you that hospitals do deal in medicine but so do pharmacies. The ability to store vaccine you are probably more right about hospitals.