I was having the same problem for about the first 3 days of turning on my 13 Pro then all the sudden I woke up and noticed the notifications were suddenly working correctly again. I have no idea what occurred to make notifications start working correctly again.
It made no sense to get a notification, long process it only to get no options like 'Run" or "View Event" etc. It would only just pop open the App as though I just tapped the app to open it for the first time.
I didn't make any changes to anything on the phone. I upgraded from the 11 Pro and before transferring everything over to the new 13 everything was working correctly.
I've actually been having numerous issues with notifications in iOS 15. HomeKit is just 1 in a list of many. Rather than spending hours trying to figure it out I've decided to wait for an iOS bug fix release and see what fixes itself.