So, my mental model of American conservatism (I'm not American) was always that they were _relatively_ thick-skinned as right-wingers go; George W took a fair bit of public criticism, say, without seeming to get particularly upset about it. Trump didn't react well to criticism, obviously, but he's kind of a baby, and a bit of a special case; extremely privileged and not a career politician. I've got to say I'm kind of surprised that the rank and file are so sensitive; I kind of thought this was largely _gone_ in western democracy. Feels very, very old-fashioned (or, as mentioned, Soviet).
I sometimes wonder if I'm missing something fundamental about how Americans think. This behaviour would be interpreted (beyond being anti-democratic, but parts of the right wing won't necessarily care about that) as showing extreme weakness here. Like, to the extent that a politician wouldn't do it, even if they _could_, because would be very politically damaging; "my opponent is a weird baby" is a stupid talking point to hand to your opponent.