Solved it!
Short version: Manually updating to 33.0220.0220.89 worked, and OTA pulled further updates after .89 was installed.
Long version:
I got back from support a couple of days after opening this thread. They said my problem was hardware-related and that I should RMA my phone. Finding that odd, I requested the problem to be escalated internally. A couple of days later, I got a second "confirmation" of the same diagnosis. Just a heads-up for those struggling with similar problems... support don't always know what they're talking about.
I had almost given up when @Mansi_ASUS (huge kudos!) linked me version 33.0220.0220.89 update file. Upon copying it to Internal Storage base folder and restarting, the phone recognized the update file - same as it did before with versions from Asus website - but this time the update succeeded.
After that, System Update ceased saying "Your system is up-to-date" and managed to detect, download and install multiple updates. I am now happily running 34.1004.0204.180. 🙂
