If I am using an HD Homerun, will the live OTA channels show up in the "Live" tab of Google TVs or will I still need to use the app "Live Channels? Or... Will the TV button/app built into the TV allow me to import the channels from the network tuner?
You will need to use live channels app or plex or the hdhomerun app.
When your TV is on the same network as the HD HomeRun (HDHR) all HDHR channels show as a DLNA device and no apps are needed. This is awkward, but works. Don't know about Google TV, but on my set top Android box I run Kodi and use the Zuki addon for OTA TV
https://github.com/djp952/pvr.hdhomerundvr/wiki
For $35 a year you get a full TV guide.
Only YouTube TV, Sling TV, Philo & Pluto TV will show up under the "Live" tab.
"Live Channels" app has been removed as a system app from Android TV 10+ devices, including Google TV devices. You need to download it from play store.