All they had to do is be a little witholding and say that vaccines were prioritized for the very sick and very important people and maybe leak some important asshole in a private conversation saying it was a "good thing" that most people wouldn't be getting it. Just do what the French did to popularize potatoes... post guards on your potato fields and tell people they can't have them.
Instead they took the line of "we know better than you and we're going to force you to do this".
Democrats have a real problem with the "we know what's good for you" attitude.