Apple Home/HomeKit became a mess for me with 16.1, it actually worked using 16.0 for me. All of my automation stopped working. Didn't matter if they are location-based or simply time-based. All stopped working.
Normally it was always our bedroom Apple TV (4K Gen.2) that was listed as the "Connected" Home Hub in Apple Home. After updating to iOS 16.1, tvOS 16.1 and audioOS 16.1 I noticed it was our OG HomePod in the Kitchen that was listed as the "Connected" Home Hub.
There is no way to enforce anything within the Home-app, there is no way to remove devices as possible Home Hubs either. We have 2x Apple TVs (both 4K Gen.2) and six OG HomePods (now down to five as one got bricked when trying to install audioOS 16.1...) so the only way I was able to force one of our Apple TVs back to being the connected home hub was to reboot all our OG HomePods in quick session so it was only our two Apple TVs that was available, only then would our Apple TV located in the living-room take over as the Home Hub and all our automation has worked since.
For good measure, I forced it back to being one of our OG HomePods and that resulted in automation not working again. Seems like these OG HomePods are terrible as Home Hubs.. Makes it extremely annoying to not have any control over what device is the preferred home hub.
16.1, and looking at reports from the first 16.2 BETAs seems to point at Apple Home pushing further into using Matter and the OG HomePod does not support Thread which is a huge part of Matter. Hopefully, this means 16.2 will abandon the OG HomePod as a Home Hub otherwise this will get even more frustrating if key features for fully supporting Matter is going to be the luck of the dice depending on whether one of our five OG HomePods or one of our two Apple TVs that fully supports Matter + Thread is acting the connected home hub.
EDIT:
Just to confirm. It was the same for me as dmfresco is describing in terms of pushing the "Test Automation" button in the Home-app. Testing and triggering using Siri was working just fine. But neither location-based or time-based triggers was working.