New vet verdict: his problem is communication, not necessarily skill (though he’s not very good at handling the animal). The operation was probably necessary. It’s hard to judge as a non-expert, which is why we’ve had some really bad experiences with doctors (remember my teeth?), but it looks like this disaster was unavoidable.
Cat is recovering well so far, which is at least something.
(Edited again to add: I just realized I deleted the original text of this post, in which I complained at the universe.)