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.
The vaccine was developed under Warp Speed, a Trump program (one of the very few things I liked from Trump). It was first made available while Trump was still president. I have some empathy for people who think it was oversold but that wasn't solely a Democrat problem.
The vaccine was mostly developed before the pandemic and then was designed in a day or 2 after the virus was isolated and characterized. Then they started testing it for safety and effectiveness.
Warp Speed was about being ready to make lots and lots of it once the testing was complete. It did a lot of good, but they probably should have spent more on it.
How so? The US COVID vaccination response was nothing short of incredible. Even with the knuckledraggers spewing nonesense, the vaccine was ready and widely available in the United States incredibly quickly relative to most of the rest of the world. I had completed my vaccination series (as a in-good-health, 20-something) before my Canadian friends even had the opportunity.
The next level up from the Democrats objectively horrible attitude is fascism, which we are currently attempting. I believe Americans have a strong cultural preference to be extreme bullies whenever they can get away with it. So often we are just execrable people who care nothing about others, at all.
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.