Hi,
Good that you set the timing.
I would look at the spark plugs, they need to be clean so the engine will run.
If the bottom end of them look wet with gas, dry them.
If they are oily, remove the oil.
If the contacts are black, use sandpaper and remove the black, and shine them up.
Check the gaps, they should be .023".
Spark plugs can quit, and not fire, then the cylinder won't fire.
The starter drive is probably working normally, it is supposed to kick out when a cylinder fires.
If other cylinders are not firing, the engine won't run.
If the points haven't been serviced lately, I would look at them, and file or replace them if needed.
TM Tractor has new points.
The Cub owner's manual shows how to get to the points, and has pics of them in the magneto.
I would charge the battery fully, use a small output battery charger, and let it charge slowly.
A 6 amp charger is good, one that automatically slows the charge rate as the battery charges up.
It is normal for the carburetor to drip out the bottom drain hole if the engine doesn't run after running the starter.