Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
So after some tinkering I have finally managed to figure out what was wrong. I'm posting the steps I took to fix this issue in case they help anyone else out in the future.
1: When logged into my online Microsoft 365 account, I followed the steps to set up an application password.
2: I closed out all Microsoft Office applications on my computer. I opened Word and selected the "Use Office to view your files for free" option. This let me open files without editing them, but more importantly, after a couple of tries* it opened an alert informing me that there was a problem with the MS Office database, typically related to having restored from a Time Machine backup.
*(at first, strangely, it alerted me that there wasn't enough disk space to run MS Word, even though I have 8GB of RAM and 75GB of free space on the HD)
3: I followed the prompt of that alert, which took me to the Office Database Utility and allowed me to rebuild the database from the restored backup version. After taking this step, I closed out the applications and opened the Microsoft Office Setup Assistant (trying to activate the application within Word or Excel still froze the application).
4: I selected the "Sign into an existing Office 365 subscription" option. It still gave me the 'temporary server issue' error message, but this time the "Try again" button was responsive. I clicked that a few times (it may take you several clicks to get through) until it took me to an email entry field. I entered my email. I couldn't click the interface's button to enter, but hitting the enter/return key on my keyboard worked.
5: When prompted for a login password, I entered the application password I'd set up earlier. Again hitting the enter key was more effective than trying to click the button.
6: I closed MS Office Setup Assistant and opened up Word to find that everything was set up successfully and working fine.
So in summary, the reason for my problem was the fact that I had restored my computer from a Time Machine backup, and the basic solution for my problem was rebuilding the software database using the database utility. The reason this was so hard to figure out was a) the error message I received about 'server issues' did not even hint at the problem that was actually going on, and b) getting the helpful prompt about database issues required somehow getting past the login/setup screen to the actual application, which seemed impossible at first with the login issue I was experiencing. I'm not sure if setting up the application password was necessary to resolving the issue, or if it only required that password over my normal account password because I'd set it up, but since it was a part of the overall series of steps I followed to resolve the issue, I included it in the instructions above.
I really hope this helps anyone who finds themselves in a similar predicament in the future. This was a tricky issue to resolve, and the lack of clear-cut answers online on forums or from customer support was frustrating.