HA will happily run offline; if you mean HAOS then I don't know what it does but it's an unorthodox Linux distro, but once it installs it should also run offline without issues. I'm also using their skyconnect zigbee coordinator and it works very well.
Yeah one of the tests was a RPi image and it wouldnt complete without a LAN internet connection (only got 4G). And it seemed far too weighty for a bit of home automation.
I recall the online requirement was for some ntp server requests that cant be disabled.
Yeah that's more of a rpi hardware requirement as it doesn't have a battery and you realistically want to have accurate time on your smart home controller, even - especially - after it cold boots after power loss.
