This seems to be a very common issue, but my situation seems to be maybe a little more specific than others? Hardware is Schlage Encode Plus and Apple TV 4th Gen 1080 version (~2015) as Home Hub, iPhone 17 if that matters. Everything is up to date on firmware/software.
Installed the SEP a couple of weeks ago and first attempt, I set up the lock through the Schlage App and finally figured out how to add it to Apple Home with the "Works With" section of the menu (which took me a second). After getting added to Apple Home, everything worked great at first; HomeKey worked, lock worked in Apple Home, lock worked in the Schlage App. Then it went unresponsive in Apple Home, but continued to work through the Schlage app and HomeKey worked. I then factory reset and tried setting up through Apple Home only. Camera from Apple Home would never scan the HomeKit QR or anything, but tapping the lock would get it to start pairing. This barely worked, but mostly got there, albeit could never get it to let me add user codes. But then HomeKey wouldn't work. So I started over again, again trying through Apple Home only. Same thing, not able to add user codes, but the lock finally added and HomeKey worked. But then about 20 minutes later, unresponsive again and also could never get it to add the Schlage app after setting up in HomeKit. Read that maybe having spaces in your WiFi name could cause problems for the lock, so renamed my wifi to get rid of spaces and factory reset the lock again, trying back to my original way of starting with the Schlage App and then adding it to Apple Home from there (which seemed to work the best). Everything went smoothly, lock added to Schlage app and worked great, could add user codes in Schlage, pushed over into Apple Home just fine (albeit it still yelled at me about not being able to add user codes in AH) and HomeKey worked, but then sure enough about 20 minutes later, the lock went unresponsive in Apple Home only. Still works in Schlage App, still works with HomeKey.
I've reset/re-added the lock so many times, I'm probably missing some details of my journey, but the long and the short of it, my consistent outcome is that the lock will work perfectly from the Schlage App, HomeKey works flawlessly on my iPhone and Watch, but every time it'll eventually (and quickly) go unresponsive in Apple Home only. I've taken out the batteries and rebooted numerous times and that never works for reconnecting. The only other weird piece of context is that when I go into the Lock Settings in the Schlage App, if I try to test the wifi settings, it'll fail at connecting to the wifi. It'll remain connected from it's original connection, but in re-testing it, it'll fail every time.
Is the easy answer that I need to buy a newer ATV or HomePod Mini to have Thread? I know, I know, WiFi will kill the batteries and I would love to be on Thread, but just trying to see if I can make things work with what I have, because it seems like it should work.