I was not enough fast, after a restart it looks thing went better, even if I had Bluetooth related errors.
2024-04-04 01:47:36.346 WARNING (MainThread) [bluetooth_auto_recovery.recover] Could not reset the power state of the Bluetooth adapter hci0 [00:1A:7D:DA:71:15] due to timeout after 5 seconds
2024-04-04 01:47:36.350 WARNING (MainThread) [bluetooth_auto_recovery.recover] Could not cycle the Bluetooth adapter hci0 [00:1A:7D:DA:71:15]: [Errno 1] Operation not permitted
2024-04-04 01:47:36.352 WARNING (MainThread) [bluetooth_auto_recovery.recover] Bluetooth management socket connection lost: [Errno 22] Invalid argument
2024-04-04 01:47:41.351 WARNING (MainThread) [bluetooth_auto_recovery.recover] Bluetooth adapter hci0 [00:1A:7D:DA:71:15] could not be reset:
2024-04-04 01:47:41.357 WARNING (MainThread) [bluetooth_auto_recovery.recover] Closing Bluetooth adapter hci0 [00:1A:7D:DA:71:15] failed: [Errno 9] Bad file descriptor
2024-04-04 01:49:27.251 WARNING (MainThread) [switchbot.devices.device] None (C0:5A:8A:D2:ED:CF): Error disconnecting: [org.freedesktop.DBus.Error.NoReply] Remote peer disconnected; RSSI: -47
2024-04-04 01:51:31.301 ERROR (MainThread) [habluetooth.scanner] hci0 (00:1A:7D:DA:71:15): Error stopping scanner: [org.bluez.Error.DoesNotExist] Does Not Exist
I restarted bluetooth.service then reloaded the Bluetooth integration and no other error until now.
I did not have these errors before when starting Home Assistant.
For the information, I use passive scan.
Until now some thermometers have no battery information available.
EDIT: IIRC battery often takes some time to be displayed correctly, except that it looks to be OK until now.
First start after update has been a little bit chaotic but second start looks OK.