LiquidFractal's post in WEIRD: "PHP sessions not working on your server." was marked as the answer
OK, update: I opened up a new browser which happened to have my test site as a default tab. So I thought what the hell and signed in to the adminCP. Long story short, I was able to access the upgrade, which completed successfully. So now I'm a bit confused since the "PHP sessions" error didn't appear to have anything to do with the VPS, but rather with issues re: browser caching. At any rate, the test site successfully upgraded (although my confusion hasn't 😆).
LiquidFractal's post in locked out of test installation was marked as the answer
Well, it's a bit of an exotic issue...I think it was in fact a browser/caching issue or something to do with what I'm currently troubleshooting, which involves https: connections. So I had edited conf_global to toggle SSL on/off and it did weird things to my login, and then when I switched it back my browser (Edge) wasn't playing nice. So I switched to another browser and that seemed to clear things up.
So, long story short - let sleeping conf_global.php's lie. 😉
LiquidFractal's post in Some can't see "Drag files here to attach..." field was marked as the answer
Ahh, so. Some of my clients had been switched to another member group and this setting hadn't been adjusted for the destination group. They had been able to use it before and it had disappeared, so this explains it.
Thanks to everyone for their help!