You can't start the connection from Hubitat. You have to start the connection from Google Home. So, I don't understand what you mean when you say:
You shouldn't have to add the Google Home app in Hubitat, it should already be there. If it isn't, then the only way to get it back is to go into Google Home, unlink Hubitat and add it again. Adding the app manually will not link it with your Google Account.
When you say "Hubitat is not connected anymore" are you saying that it was at one point and you had devices synced to Google Home? Then the Google Home app in Hubitat had to be installed. And apps can't uninstall themselves, so you had to have removed it. If you did, you can't just load a new instance of it. You have to start over with the Google connection.
If you’re still having issues linking Google Home to Hubitat, you may need to follow this procedure to remove any remnants within your Google account...
The Google integration is enabled - BUT I have some devices (Aqara Window sensor and Aqara Temperature / humidity sensor) which don't appear in Google Home - they work in the dashboard but are not passed on to Google Home. I might open a new topic to ask for that issue.
You may may to look at the following thread. (Note: I have not used this Community-Maintained Google Home integration, so I have no first-hand experience.)