If a corresponding device is connected in Apple Home and you want it in Home Assistant, you just need to remove it from Apple Home, but not reset it. So it remains connected to WiFi and can be found by Home Assistant. These devices were now displayed in HA without searching after I removed them from AH.
With a HomeKit Bridge you can make the device available for Apple Home again, so that it can be switched on and off via the Apple app.
HomeKit devices cannot be paired to more than one controller (hence the “no unpaired devices could be found” error). So you have to remove them from the Home app on iOS, possibly reset them (device-dependent), then add them to HA.