Final update (hopefully): A few weeks ago, Apple finally found the problem that’s been affecting me and at least hundreds of others. It was a bug on their servers that affected some accounts. It appeared to be totally random; didn’t follow any patterns (types of accounts, etc.). Anyway, I hope they can fix future issues in a shorter time than ~3 months. I’m happy to have my iOS devices working again.
Thanks to Nate at Apple for sticking with me over the past month and providing such good customer service.