Solved another problem with the new WordPress implementation – permalinks are now functional again. Everything I found only kept pointing me to reset the .htaccess file, which functionally was irrelevant (but likely viable in a different case.
For my issue, the specific problem was that pretty permalinks was broken, but basic permalinks worked fine.
The solution was actually an Apache issue – when I enabled SSL, I used a bare bones configuration for the sites-available file. In my case, I needed to add the <Directory> directive to the SSL virtual host (it existed only in the non-SSL virtual host. Specifically, what I added was:
<Directory /var/www/(sitename)/>
AllowOverride All
</Directory>
Once that was done, a quick refresh of Apache, and pretty permalinks was functional again. Hopefully this may be helpful for someone else, or simply as a record for myself for the next time this comes up.