Why would she say anything at all then? It would be impossible for Trump alone to authorize and recommend the vaccines. To many Democrats at the time this statement sowed doubts about the vaccines. My parents said they wouldn't get "Trump's vaccine". What you are failing to acknowledge is that how you understand the statement is not how everyone would interpret it, and many interpreted it as being against the vaccines because Trump was "rushing" them.
I don't hear this being brought up by Democrats at all. And if you look at the statistics by party on who is getting vaccinated I think it is pretty stark that this is not having much of an effect. So even if that caused some Democrats to think that at the time, they seem to have gotten over that.
Looking at the both the statistics, and what media sources are pushing anti-vax sentiments, I don't think your argument has much of a leg to stand on. It is distorting a statement to try and justify a "they are bad", or a "they are bad too" attitude.
