Homekit, ATV4 HD hub, mix of accessories, and "No Response"

My home was absolutely Homekit-happy with eleven accessories all working well. Suddenly, everything went to "Not Responding" at the same time.

This was just after I had bought a new iPhone and migrated everything from an older iPhone to it, but I have no other indications that this had anything to do with this problem.

I checked the hub, an Apple TV 4 HD, and found an update waiting for it (14.5). I installed the update and the TV restarted. No improvement.

I used Airport Utility to confirm no updates were pending for my router and bridges, and then restarted them all to be sure.

I signed out of iCloud on the TV and signed back in. No improvement. But at this point TV prompted me to "Finish" the Homekit setup by placing the new iPhone in Bluetooth proximity with the TV. This had not appeared before and I think it must be provoked with a sign-off.

This only changed the TV's accessory status from "Not Responding" to "Needs Repair". Everything else remains "No Response".

I could find no guidance (none) on the web about what to do about this Homekit status, but while I was searching (ten minutes?), the hub (the TV) cleared that status on it own, and now responds. But I'm back to square one: the other accessories are all "No Response", still.

The accessories are from elgato (Eve) and Lutron (Caseta, with bridge).

I think I'll just remove all accessories, remove the hub, reboot everything I can reboot, pull and put back the batteries in the others, and rebuild the Home from scratch.

I would love to have some solid information on how this can occur (e.g. what "not to do") and how to more scientifically resolve it if it recurs. Thanks for any replies.

(Setting 'Topic' to 'HomePod' as there is no specific 'Homekit' topic in my choices)