I would probably try to engage with iTunes Support if this was the first time I've seen this issue (or if it was specific to my account only – which I'm almost positive it isn't). It seems to affect only a small number of users, apparently not enough to cause mass complaints or get any media coverage. I had a hard time finding any relevant information about this problem when I first encountered it a couple of months ago. It is also something that seems to get broken and than get silently fixed a few days later on Apple's side.
So my strategy for now is to try and wait it out. I don't have enough patience to jump through all the standard Apple Support hoops (signing out and in, restarting and hard resetting my devices, setting them up as new ones, etc.) just to confirm that it doesn't help. There is still a workaround if I really need to update some app – just delete or offload it first and then download the latest version from the App Store. It works this way, but regular "update" buttons don't.
Anyway, thank you for the suggestions.