Ok, time to summarize the situation. (Punchline: problem seems to be fixed! Here's how.)
Side issue: the steps to fix permissions now work with an update to the Dropbox app itself. (Namely, open Dropbox preferences, go to the Account tab, and hold the option key (on a Mac) to see the options to Fix Permissions or Fix Hardlinks. Be aware that when you click those buttons there's no visual feedback anything's happening, but if you wait a while you should get a message the operation has been successful (or presumably not, if not). That did not, however, fix my particular problem of Dropbox sync stalling after just a few files every time it restarted. But at least the option is now available on Big Sur if needed in the future.)
After being bumped to a couple of higher levels of support... [One of the lower levels, I kid you not, tried to argue that the issue I was having was because Dropbox was bad at what it does, and it was unreasonable of me to expect Dropbox to backup a lot of stuff. In spite of the fact that Dropbox tries to upsell me to a larger Dropbox every time I login. But anyway...] ... support suggested unlinking Dropbox online from the machine exhibiting the problem and then re-linking it. This worked! And it did not require re-uploading all of my stuff (somewhat less than 2TB), or an unreasonable amount of time, both of which I had feared. But since the only alternative was switching to another backup solution (my planned next step), I was willing to give it a try.
Here are the steps I took:
1) Login to Dropbox.com
2) Click on my avatar [top right] and choose Settings
3) Select the Security tab and scroll down to Devices
4) Find the device exhibiting the problem and click the trash bin icon to the far right. [Note: this was a bit scary -- I was concerned Dropbox might do something nutty like remove files on my home device. I proceeded after reading the information under "Learn more", which promises it won't do that. Still, if you're smart you'll have a complete backup before proceeding.]
5) Click "Unlink" in the resulting dialogue. [Do NOT tick the scary checkbox.]
6) Wait a few minutes while Dropbox on the affected machine spins around in confusion.
7) When a dialogue pops up asking to re-enter your account information on the affected machine, do so.
At that point Dropbox on the affected machine began furiously re-indexing my Dropbox (/not/ re-uploading everything, which I had been fearful of), and within an hour or two had completed re-indexing everything in my Dropbox, synced all the stuff that hadn't synced over the last couple of months (with the expected assortment of "conflicted copies" of some things, so be sure to check carefully which is the one you want to keep -- in my case, it was always the "conflicted copies"), and for the first time in months finally reported itself as "Up to date". It's been a few days and Dropbox has been humming along happily ever since, syncing my stuff more or less instantaneously as it always had up to the point this problem began (when I updated the machine in question to Big Sur).
So I'm going to report this as "solved" for now! Thank you for everyone's assistance. Even the nice person who said Dropbox was bad at what it does 😉
[If they want to be taken seriously -- and to keep taking people's money to provide, you know, a file backup and sync solution -- they really should stop saying that.]
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