"What is the current agent version of the affected Macs running Ventura?"

ScreenConnect_22.10.10924.8404_Release

ScreenConnect_22.9.10787.8389_Release

ScreenConnect_22.9.10446.8359_Release

"Are you issuing a reinstall command from the host page? (assuming yes here, but want to verify)"

Yes.

I also completely uninstalled vScreenConnect_22.9.10446.8359 from one Mac that I have in my office, build an installer for v22.10.11109.8417, it installed successfully, but it never showed up on the host page. Screen Recording and Accessibility were enabled under Security & Privacy just to let you know.

I still can connect to all these affected machines but can't update/reinstall the access agent. This is only happening on Mac OS Ventura, on the other Mac OS versions reinstall/update/clean install works just fine.

"When the reinstall command is issued, what happens with the agent? (Does it go offline as though its updating?)"

No.

On the machines running Mac OS Ventura nothing happens. On all the other ones, when I click reinstall it goes offline for a couples of seconds and comes back online fully updated.

I was able to get the Support Agent to work on Mac OS Ventura, but  on a regular base, I need access  to some unattended machines and that's why Access Agent is so important in this situation. I also tried "Install Access" during a Support Session but it didn't work either.