My tried-and-tested method has saved my (company’s clients) ass a few times.
Every Mysql/MariaDB server has at least one replication target. This replicant is not used for access by the infra, and can be paused, restarted, etc with no issue and is configured with this in mind.
We run a mysqldump on the replicant. Depending on the resiliency required, we store the dump on the replicant and/or a third location.
The tools differ, but the practice applies to pretty much every database system and the database has the benefit of not being interrupted during the backup (replication is paused during the backup, and resumed after completion). This also has the benefit of already having replication configured, and adding a secondary redundant instance you can swap out for the master (or using the backup replicant in a pinch) means disaster recovery is much faster.
Also, I dislike many things about Azure’s offerings, but their Flexible Database for MySQL does the above for you as one nicely packaged solution for a reasonable-but-not-cheap price.
I mean, for 10 bucks anything is a decent deal. Those specs are pretty decent for a simple home server. I’m not familiar with HP thin clients, but I assume you can install a Disdro of your choice on it? My big reason to avoid HP is their crap software and warranties, both of which are moot here.
I would say relatively light software like tailscale, pihole and such would be fine. Docker containers might be pushing it, but that depends largely on what containers you want to run, same goes for nginx; by itself the requirements are fairly low, it depends on what you want to run on it.
Jellyfin might be a stretch, and as you alluded to, real-time transcoding is probably out. It strongly depends on the decoding capabilities of that chip and wether it does hardware decoding or if it all happens in software. The latter might be too much for it. If it can handle it though, it might be interesting as a media player hooked up to a TV, rather than acting as a transcoding or DLNA-esque server.