I think this depends highly on your mouse skills. Most of the top bullet players I've seen play on stream (Andrew Tang, Hikaru Nakamura, and Daniel Naroditsky) use drag-to-move.
A notable exception is Magnus Carlsen. He uses click-to-move, but I think his skill in bullet comes from his baseline chess skill and not his ability to move fast.
on computer, the difference between the 2 is negligible in my opinion, since either way you have to place the mouse at the start and then navigate to the finish with the only difference being if you are holding down left click or not.
on phone/tablet, the difference between the 2 is massive, since you don't have to slide your finger across the screen and can just tap tap (and even use multiple fingers if you want.
On Lichess, this isn't the case; if I set my movement preference to 'click two squares', a click on a piece is registered immediately on mousedown regardless of cursor movement.
(When I set my movement preference to 'either', it's a bit harder to test, but I think a brief click-and-drag always counts as a click provided the mouseup happens within the initial square.)
The ability to physically move fast has relatively little role in bullet beyond a certain level, because most games will end on the board instead of in a frantic mouse scramble. This was even tested, to some degree, in one of chess.com speed chess championships. They had a bunch of top players including Alireza and Nakamura play one of the little reflex games where you have to click in a certain spot that lights up, and their performance was nothing noteworthy. Nakamura, the oldest player by far, was actually the fastest but his reflexes were again nothing spectacular, so it was more of a reflection of the reflexes of the other players.