Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches.

This isn't really a signaling issue so much as a response to the fact that mathematicians have had centuries of experience where apparent theorems turned out to be not proven or not even true and the failings were due to too much reliance on intuition. Classical examples of this include how in the 19th century there was about a decade long period where people thought that the Four Color Theorem was proven. Also, a lot of these sorts of issues happened in calculus before it was put on a rigorous setting in the 1850s.

There may be a signaling aspect but it is likely a small one. I'd expect more likely that mathematicians err on the side of rigor.

ETA: Another data point that suggests this isn't about signaling; I've been too a fair number of talks in which people in the audience get annoyed because they think there's too much formalism hiding some basic idea in which case they'll ask questions sometimes of the form "what's the idea behind the proof" or "what's the moral of this result?"