I've seen a couple of older Lawn-Boy shaft drive mowers where the shaft
isn't spinning. Sounds expensive, but it's really simple.
The drive is a square shaft between the engine and differential.
There are two larger square blocks spread apart by a spring. If you
clean things off enough, then the blocks will squeeze up together and you
can slip the shaft drive assembly out from between the engine and differential.
And THIS is what happens, the end by the engine gets worn ROUND and
the drive doesn't engage! The right thing to do is replace the shaft
with a new one (or at least another from some other LB. I've actually
made a temporary fix by swapping ends and then shimming the differential
side block out a little from the differential with a washer or two.
It will get a couple more lawns done!
Put it back together and mow some lawns!