I never heard of Cozy, but it looks quite nice. The Self-Hosting Documentation ist a bit lacking, but https://github.com/cozy/cozy-stack-compose contains all required information to set it it up yourself.
I originally used Nextcloud, but it has a lot of features not related to file hosting
Cozy seems to be in a similar situation, where file storage is just one of many features that it provides. If you want just files, it might be the best idea to just use any WebDAV Server or something like File Browser.
This isn’t some “idiotic principle invented by Ubuntu”, it just follows the freedesktop.org Trash specification. More many users, it can be really beneficial, see also the spec’s introduction:
If an application like Prism Launcher should use the trash can or delete the files directly is an entirely different question.