I'm (currently) on Raspberry Pi OS (as I need something that "just works"). Home Assistant is running in Docker like everything else.
A lot of apps use hard coded paths, so using a subdomain per app makes it much easier to use them all. Traefik has middleware, including stripPrefix, which allow you to strip a path prefix before forwarding the path to the app, though - have you tried that approach?
I should've mentioned this, but I'm using Nginx (I really enjoy the simplicity of just having to add a section to a file whenever I want to add something). Before running HAOS I was running RPIOS again and used Traefik, it worked (but felt like a lot more work to setup than just a plain Nginx setup).
Edit: I forgot to mention, but there are things like stripPrefix for Nginx, I'm going to look into them. Although, this is what I meant, when you start to do things that are "advanced" with Home Assistant they turn into "hacks," and the barrier for advanced things feels a lot lower than with other self-hosted services (and I get that Home Assistant is very complex under the hood, it's just frustrating).