@karls0n The charge regulator on board, TP4056, is a very good device.
The LilyGO T-Display S3 boots back up in seconds, so it should be shut down when not in use.
I would not leave a charging Li-ion battery unattended for very long, Li-ion are very touchy about re-charging too fast from empty. The TP4056 takes care of that but there are worst-case or extreme situations...
It is designed for a couple 16550 types.
It is intended that the USB supplies the charging current, so the LilyGO T-Display S3 is "ON" while charging, but you could program the LCD to turn off the backlight whilst charging!