If you have not cracked her open yet.
take a oil squirt can and inject some motor oil into the spark plug hole spin the engine over by hand couple turns then do the leak down test again and if it's the rings your 60% leakage will get to less leakage and the compression should go up also if it's the rings.
The motor oil only temporary seals the rings blowby. Another test is to use a vacuum/pressure gauge (compound gauge 0-10 pressure, 0-30 vac, automotive type for about $15) and measure the crankcase vac/pressure. (you can review this test on-line) On a good engine the crankcase will have a slight vacuum, bad rings will produce a pressure. Just another thought you might pull off the breather hose from the carb throat and start the engine and see if you see any smoke or oil coming out of the breather tube. A leaking breather check valve should also not be overlooked. You can review it's operation and how to test on-line.
The 60% leak down test you made is a hint of internal leakage and you are now trying to determine if it's the rings and worn cylinder. You need to have the cylinder mic'ed to see if it's within specs if you are going to re-build/re-ring for yourself. New rings will not do any good for very long if the cylinder is out of spec's.
Sometimes what causes the rings to go away fast is someone when doing carb work accidently left off the breather hose at the throat of the carb and this bypasses the air filter and the engine will suck lots of dirt into the carb throat. This breather hose is in behind the carb throat intake on yours and most Briggs and Kohler small engines and is easily overlooked and not readily apparent that it's off until the engine has already sucked lots of debris into the carb throat bypassing the air filter.
Those engines are also know for head gsket leaking between the oil galley and the cylinder, but you said you replaced the head gasket.