But in practice I strongly suspect that it would be more trouble than it's worth...

Personally I would start by working out what version of WordPress your current server has and do some googling on how to back up/export the data. Then once you have it safely backed up, google migrating that data to a new server and/or upgrading to a newer version of WordPress. Ideally leave your old server as is as long as possible as then worst case scenario you can go back to that and re export if need be...

Build a new VM then transfer the Wordpress site across to the new VM. There are a number of Wordpress plugins that will do it e.g.  xcloner.

If you only have the one server run xcloner or alternate to create a backup of the wordpress site, install a newer TKLX from ISO on the server (Wordpress appliance or LAMP appliance and install wordpress) then re-install the wordpress backup plugin and do a restore.

To be safe I would probably launch a cloud VM (AWS or other) and get it working on the cloud server before trying to rebuild the one server you have. Once you're happy with the upgraded server you can kill the cloud version. Should only set you back a few bucks...the beauty of on-demand infrastructure :-)

Yes...unfortunately working Wordpress configurations can quite easily stop working properly on a theme change or a plugin change. If you need to switch themes because a theme is no longer being supported, switch to a newer theme from the same author if available. It is more likely to work than picking a theme designed by someone else.