As you have surmised, a loose valve seat, not unknown on Briggs engines that have overheated . Providing that the machine has not been run in that state for too long and the seat recess enlarged too much there is a procedure for recovering the situation, which if carried out to the letter should provide a permanent solution. The procedure is covered in the appropriate Briggs manual which is not expensive but does advise the use of one special tool, a valve seat cutter to ensure that the re-positioned seat is concentric with the valve guide. However, if the centring advice is followed exactly you can usually get away without re-cutting the seat.
The appropriate manual for your “L head” (side valve) engine is part number 270962.
EDIT. Should have watched all your video after the discovery of the loose seat! Lets just say, not by the book but sufficient for your purposes!
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