OK so after one week of struggling with the same issue on my iPhone XR (iOS 16.6), I managed to resolve it.
Here is what i did, inspired by other comments found in this thread :
1) Restart iPhone
2) Reinstall "Home Kit" and "Apple TV" apps (those two were not installed on my iphone)
3) Turn off iCloud temporarily (Settings >> iCloud >> Log Out)
(this is a very important step ! Since my HomeKit app was stuck on the fisrt screen "finding / loading scene" or something like that...sorry I don't remember exactly)
4) In the Apple TV settings, under "Allow TV to Access", turn off "Allow Live Activities" and "More Frequent Updates" and close the app.
5) Open the Home Kit app. On its main page, tap "..." at the upper right corner => "Home settings" => slide to the bottom of the page and tap on the "Delete home" button. Close the app.
6) Turn iCloud back on
Hope it can help !