I don't think it was the roll-out or anything along those lines, so much as the lying. Saying people won't get sick or won't catch COVID if they get the vaccines [1] was just false and very myopic. Obviously the intent was to try to get people to take the shots, but they completely destroyed trust in the process. Broader skepticism predictably follows from there.

[1] - https://xcancel.com/ITGuy1959/status/1581034815700488192