I don't think that's an entirely accurate presentation. I think the Democrats supported a vaccine roll-out but stated that they did not trust the Trump White House to lead it, because the White House had proven in the past to put politics ahead of science. Trump had been pushing for the vaccine to be made available by October, which he called "a very special date" and it was obvious that Trump was attempting to have the vaccine available before the election so that he could claim it as a success. The Biden campaign countered this obvious attempt at election politics by saying that Americans shouldn't trust Trump about vaccines, they should trust scientists. Biden specifically said "I trust science, I trust vaccines -but I don't trust Donald Trump."
But these dynamics were political rhetoric and I don't think it controlled how anyone actually thought and behaved about the vaccine.
I think if you step back and look at the constituencies, there was always going to be a solid block of Americans who mistrusted the vaccine and loathed vaccine mandates that fit within their worldview - and those were largely going to be on the right/Republican spectrum of American politics, because there is a fairly strong libertarian, anti-science, anti-institutionalism, anti-academia trend in that constituency. The anti-vax movement was already a part of that constituency and Covid only accelerated that marriage. I don't think Trump's late-hour support of the vaccine had much impact on that behavior.
And at the same time, you have a constituency on the left that have absolute, perhaps even unfounded faith in institutions and academia that were always going to be supportive of a vaccine, when it was ready - and they would be willing to take their word for it when that was, as long as the word was from a trusted source . . . not Donald Trump or Donald Trump's health messengers that were highly suspect. But when the vaccine was actually available from reputable sources, I think that was all they needed.
Certainly I don't mean to say that everyone behaves in one bloc or another. Plenty of MAGA populist Republicans got the vaccine because they trusted the medical science and plenty of Democrats felt uneasy about a fairly new technology and decided to trust their immune system without enhancement. But i think the broader trends were very real: vaccination rates were much lower in red districts and much higher in blue districts. And rates of severe disease and death after the vaccine roll-out were much higher in red districts and much lower in blue districts.