Update.
The short version: The backup was corrupt. New backups remained corrupt regardless of backup to Mac or to iCloud. That’s what prevented syncing. It also explains why syncing on the new device was broken (as long as that device was set up via restore from backup). Solution: new iPhone had to be set up w/o restore from backup. Tons of stuff was lost and had to be reinstalled and reset by hand. Despite biting the bullet of iCloud syncing. A real PITA. But now she has a new 128 GB iPhone SE that has most of her stuff, runs well, and syncs. 
Lessons to learn: if you do anything outside of the strict Apple sphere, i.e, you don’t want to pay monthly fees for extra iCloud storage (after having already paid hundreds of $ for that storage in your iDevice) and you prefer buying your music wherever instead of renting through some Apple all-you-can-eat gig for $100/year, if you are that person, and you do not have both syncing and backing up 100% secured (it’s a good idea to take an older iDevice and see if you can actually fully restore from the backups you think you have), then YOUR DATA IS NOT SAFE. Do not assume a backup will allow you to back up to a new device. Do not assume being able to sync means you will be able to recover to a new device. And don’t think using iCloud instead of Mac is buying you any added safety.
The long version (for those with lots of patience or those who like gawping). 
We were too busy to make the Tue appointment and rescheduled for Sat. Arrived at the Apple Store at 4pm. Left after 8pm. We went through a half dozen geniuses, incl. getting advice from more experienced techs from back of the house. They were able to recreate the issue. They were able to confirm both iPhones were affected. They also tested USB vs. wifi and different ports. But they were able to confirm it was not the Mac or iTunes since they were able to backup and sync one of their iPhones just fine. At this point they believed it was either her Apple ID or DRM. DRM was ruled out because they recreated the same issue despite temporarily getting rid of all music and using just a few tracks of a freshly ripped CD (was amazed my wife’s built-in DVD drive still worked, she hadn’t used it in probably 6 years). Apple ID was ruled out after checking everything was fine there. At this point five geniuses went into a huddle and started going over everything they’ve tried and trying to figure out what the one common denominator was. That’s when it became clear it was the backup itself. That was the only thing that linked both her iPhones and was not involved when backing up and syncing to their support iPhone. Unfortunately, further testing showed you cannot just delete all backups, do a fresh backup, and that one will then not be tainted. The suspicion was that there was something on her old iPhone corrupting the backup which then led to a loss of sync. Unfortunately, whatever this was prevailed through both iCloud and Mac backups so there was no way she was going to be able to use backing up. At that point she realized she could either stick to her old iPhone which had all her stuff, but would not sync (and hence would offer no data safety in case that device ever died or got lost/destroyed) or she could move to the new device without backing up which meant using iCloud against her wishes to sync a ton of material and then aving to redownload all her apps and set up each and every system and app setting from scratch as far as possible. Of course some she will not regain at all (for example wifi passwords of networks she won’t see again until months from now). At that point the store was closing and we decided to leave. She needed time to think and we all needed a break. We went for Mexican food, I had too many beers to get myself over this colossal Apple CF, and then we calmed down and decided what the plan was going to be. She decided to bite the bullet and turn on iCloud syncing to get at least some of her stuff transferred. She set up the new iPhone as a brand new device from scratch. She got Contacts and Calendars through iCloud syncing (she couldn’t have gotten those off her Mac since what was on her Mac was way outdated because her old iPhone had no longer synced back to her Mac). That’s how she also got her Camera Roll pics. She started downloading all her 3rd party apps again. And then painstakingly started setting up all prefs by hand. There’s a gazillion in Settings, but also each 3rd party app needed to be set up from scratch. A day later and she’s still not done. Her next phone will probably not be from Apple. The new device will now sync with her Mac so it spent several hours downloading all her music and her old photos from her Mac. Books, movies, TV shows, and audiobooks came onto the phone the very same way. There’s a ton of stuff that went missing. Things like her Notes (iPhone account). There was no way she was going to use iCloud to sync her encrypted notes. Fortunately, AirDrop helped with some of that. The rest was typing off one screen and onto another iPhone by hand. For hours. She has a lot of work apps that need to be set up from scratch, but once set up will sync over their own clouds and that stuff worked. It was just a pain because most don’t just need user/pass, they want 2FA via some quirky thing like either an additional app, or a fob (requiring her security card) or some other shenanigans that makes you wonder how hard can it be to get single sign-on these days. Mail was cool once she set up her new accounts thanks to IMAP. There was tons of small stuff I never would have thought about like all your alarms, with their melodies and other settings. Bottom line, a few more days and she’ll have it all. Then she’ll be able to shut off the old iPhone and she’ll have migrated for good. At that point she will stop iCloud syncing of various apps and she’ll be syncing exclusively to her Mac again. Backups are already now exclusively Mac, iCloud backup is off and iCloud backups have all been deleted. She asked me what means we have of checking if that stuff was actually really removed from 3rd-party servers. She knew the answer. Her new iPhone now syncs fine and backups appear ok. If we had yet another iPhone lying around it would probably be a good idea to periodically check if she can back up to a new device from the sync/backup data stored on her Mac.
Getting back to Apple, I have to say Apple’s phone support and the people at the Apple store were great. They tried really hard, they didn’t treat her like an idiot, and they never tried to get rid of us despite the problem persisting over hours. Chasing down her issue must have cost Apple way more money than they ever made off that SE. That said, ultimately though, they weren’t able to fix the issue. And I’m not at all certain this will find its way back to the right Apple engineering team even though it’s quite clear there’s something that causes corruption and there’s something that lets that corruption get into the way of syncing. At least one of those issues should be fixed by their engineers and I’m highly doubtful all this info will get back to the right people in the right form. Plus, I hestitate to believe Apple really gives fixing such bugs high priority. I’ll be happy to be proven wrong. Her old iPhone is still around. Next iOS and iTunes update, you can be sure that’s the first thing I’m testing.
On macOS you can check the FS integrity and you could attempt to move/remove/restore individual files to find out the culprit, but on iOS there is no equivalent (not that I’m trying to advocate full FS access on an iPhone!).
I have no problem believing that ultimately, you’re right. I do have to admit I do not usually do that. In fact, I go through a lot of trouble to prevent having to do that just because it’s such a hassle. I’m very conservative with what I install, try to keep track of exactly what gets put where, which updates get done and where they go. On my personal systems, I try to run a very tight ship. Fortunately, my Macs and iOS devices have usually done quite well this way despite all the delta updating. But I imagine you’re right and ultimately, it’s a question of time until I’ll be forced to do what you describe.
;)