The leak-down test is not to see if there is a ring or valve problem, it was just intended as an additional check on the head gasket. You are not likely to have a ring or valve problem unless the engine has been dusted due to the air cleaner problem. Also, if your compression pressure tests at 130 psi, it sounds as if your decompressor is probably not working. Does the starter motor have a tough time getting the engine past compression, so it cranks in a series of fast-slow cycles rather than steadily?
An electric fuel pump will not fix either a blown head gasket or a dusted engine, but it will fix the oil spraying out of the fuel pump, and there is a good chance it will stop the engine from hunting when it is idling.
I can offer two alternative approaches you might consider. The first is to aim to make your engine run properly without changing from the original Briggs design. That would mean replacing the head gasket, and inspecting the cylinder bore while you are doing it, to see if there are signs of longitudinal scratches in the bore due to dust entry. If the only cause of the fuel pump oil problem was a leaky head gasket, that might fix the problem.
The second alternative is to replace the crankcase-pulse-operated fuel pump with the current electric one, so both the oil spray and the possibly erratic fuel delivery are eliminated without addressing the head gasket issue. This amounts to saying that the compression is good enough, so the gasket isn't leaking enough to matter, if it is leaking at all.
I suspect Briggs would recommend the second approach - they may even have a service bulletin issued to their dealers on the subject.
Whichever of those approaches you choose, I think you need to look very carefully at the tappet adjustment: I can't believe the decompressor is working properly, with 130 psi cylinder pressure at cranking speed.
