New old-stock carb kits are usually available quite easily from the US.
The old Briggs flo-jet carb can be very temperamental, make sure that every joint is well sealed, check that you haven’t got any play in the throttle butterfly shaft. If you do get a carb kit, run a tap down the jet holes before replacing the jets to make sure that the jets seat properly, the threads can wear with time.
All the above can allow excess air into the carb which will stop the engine running.
Spare carbs do turn up quite regularly, so it may be worth collecting a few, I ended up with a one built out of two or three different ones when I had a Trusty with the Briggs zz.

Have you got the part number for the kit?