Pretty funny to see Simpson's name showing up here. I have kept two souvenirs from my academic life: the original manuscript of my thesis and Simpson's very eloquent and overwhelmingly positive review of it.

Regarding the topic, I'd say that mathematics, as a whole, is a set of intuitions and ideas, rather than strict proofs. I've spent a fair amount of time working my way through some of Kontsevich's ideas and attended many of his talks -- he rarely (if ever) gives any proofs, my educated guess is that he proposes some extremely profound ideas and his collaborators grind through the details (not always). And it still works just fine.

I remember Kenji Fukaya saying once, talking about a PDE-heavy theorem: "We are trying to keep the details of the proof under 500 pages and it's not sa easy". The main issue here is that people who don't do professional, academic research in mathematics are unaware of the complexity of modern maths. It has evolved immensely during last 50 years, the theories are just layers and layers of foundational work one has to assimilate before getting any work done. Rigorous verification takes years and no one in the academia is being offered a job for proof-reading of existing papers.

One should also remember that referees of the papers are not being paid, it's considered "work for the community" and it's hard to blame them for not reading the papers in detail.

Mathematics are anti-fragile.