Oil enters the cylinder either past the rings , past the valves or at the very worse case through a crack in the head which includes blown head gaskets
The head gasket is alloy and rarely blows but you test this by removing the dip stick after running for 10 minutes /
IF smoke rises then the gasket is gone .
To test for a crak in the head or a worn valve seal or slipping valve guide, remove the rocker cover.
This relieves any pressure on the oil in the head to force it into the cylinder
So if it stops burning oil when you do a test run, no rocker cover ( it will be very messy ) then suspect the head .
However if your mower has the right angle rubber elbow connecting the air filter, I would suspect that the rings are gone.
Had one earlier this year , both cylinders glassy smooth
They got a quick hone, new set of rings, problem solved
This customer has a bagger on the mower and it was sucking a mile of fine dust from the bagger into the cylinder .
remove the rubber elbow , put your damp finger into the carb throat and wipe around .
If it comes out with dust on it then the rings / bore will most likely be your culprit .
The elbow is a big problem as it does not seal particularly well and is quite difficult to fit properly .
FWIW the customer found a Donaldson filter 2nd hand which ended the fine dust problem.