For everyone following this thread: in the end, I migrated all of my WordPress sites (and 2 HTML sites) by setting up a new CloudPanel server on Debian 12 (Hetzner). The transition was relatively painless; it took me an entire night plus a few hours in the morning, for a total of 12 hours.
CloudPanel is very lightweight, and I only had a bit of trouble getting CloudFlare DNS and CloudPanel's Let's Encrypt to work together properly (I had to go without the proxy mode, but I’ll work on restoring it in the future). CloudPanel is very lightweight and has everything I need except email functionality.