Not sure about that last part. But it is certainly the case that much -- not all, but much -- right-wing antipathy towards vaccines would have never come into being. Some would have still existed -- the conspiratorial anti-vaxx wing of the conservative movement predates COVID and would not have been squelched entirely no matter what Trump did -- but it wouldn't have been amplified endlessly on Fox, nor would it have become a tribal identifier for true Trumpist loyalty.
Would Democrats have simply flipped and become the new anti-vaxx party? Unlikely. We'd still see anti-vaxx sentiment from these sorts of "progressives". And they'd be roundly thought of, and presented as by other liberals as idiots worthy of contempt and scorn (though they'd no doubt be defended vociferously by Glenn Greenwald types). Hippy-dippy anti-vaxx sentiment on the left would have remained a joke, just as it had been for years before it became a conservative domain and suddenly had to be respected as a grave matter of conscience and a deep policy dilemma. The vast majority of Democrats would still get vaccinated, because at least in this domain polarization really is asymmetric and Democrats aren't willing to enroll in a death cult just to do the opposite of whatever Trump does.
Oh, and I also strongly suspect that if Trump had taking this loud pro-vaccine stance from the beginning, there's a solid chance he'd still be President today. So take from that what you will.