Does that HassIO operating system give the ability to update the firmware on a Raspberry Pi, the way the Raspberry Pi OS does, or do I need to boot off a Raspberry Pi OS card to do that.
The Pi 4 has had a number of firmware udpates since release to help keep it cool.
HassOS updates (for example see the 4.14 changelog) come with bug fixes and kernel updates, so you’re fine.
Kernel yes, but firmware no. You’ll be a second card, or you’ll need to backup, nuke the assistant setup and flash raspberry pi OS, update firmware then go back and reinstall and restore assistant.
I don’t use a RasPi, so I’m not sure of the terminology, but is the Firmware the same as EEPROM?
If so, then yes, HAOS has the ability to upgrade this:
However it looks like it’s been disabled for the time-being due to compatability issues and your instruction are correct that the user should do this themselves if needed (and if it won’t break anything):