Correct, the discovery process where you add them doesn’t actually involve the addon and works fine without it. The addon is just a wrapper for the standalone ESPhome web UI. It’s also not the only way to author and flash a config to a device, you can do it from any computer and on the command line.
It’s not needed, and you can run it separately standalone if you do want it as well.
I used both, I ended up settling on searxng because Whoogle seemed to be unable to retain my settings. Might be something with my cookie configuration, but searxng has no problem remembering my preferences. If that is not a problem for you then they are comparable; Whoogle is pretty simple to get going and works well, searxng is slightly more complicated to set up (but not that much with docker) but has a ton more features.
I’ve been using Debian as my main server OS for a long time, but yeah I’ve started using Alpine for lightweight hosts for a single app/docker lately. Especially on proxmox with the pve-helper-scripts you can spin up an Alpine container or VM with Docker in a few seconds.
High availability storage is what the Ceph integration is for.
Edit: though it’s kind of a pain to set up and probably way overkill. A separate TrueNAS or similar appliance with a 10 gig uplink will be easier and probably just as reliable for your use.
I tried to make Grocy work as a kitchen inventory and shopping tool too but damn, it tries to do way too much detail. I’m sure there are people with enough discipline that it appeals to, but I really felt like it was way too particular and entering/tracking items across areas/brands/types seemed like overkill. Its whole system seems unintuitive to me and I never could get to the point where it seemed natural, so I gave up quickly despite really wanting it to work.