lawn tractor not charging

IDK about the Cub specifically, but I have worked on hundreds if not thousands of small engines and yes the charging systems are really simple.
Usually there is a small coil mounted under the flywheel, and flying magnets on the flywheel. There are no moving parts to this so it rarely makes trouble.
This electrical generating coil creates A/C so it has to be converted to D/C, then regulated down to a voltage that won't boil your battery. All that happens inside a tiny box bolted onto the chassis, usually somewhere under or nearly under the seat, or directly bolted to the engine.
Thus, the system consists of these components;
the coil and magnets, and
the regulator, and
the battery itself.
The system works almost exactly like the one in your car.
The first thing you gotta do is charge the battery and load test it to make sure it is actually serviceable.
The second thing is to disconnect the regulator and check the output of the"alternator". You have to test for A/C, and If you get anything over 14 volts at idle, then the darn thing is working. If less than 12 well then, she's iffy, rev it up and my guess is that it might output over 30 volts at some higher rpm.
Once the voltage generator is proved to be working, it's on to the regulator.
Again, IDK specifically about the Cub;
But from my experience the regulators on most of these riders are not much more than A/C to D/C convertors; A/C goes in and D/C comes out and the battery itself is the regulator, in as much as whatever voltage comes out of that thing, it all goes into the battery. The power coming out of that generating coil is so small, that it can't hurt the battery.
So if you get D/C coming out of the regulator, it's probably good...... BUT....If you get ANY A/C at all coming out, she is broke.
This type of regulator probably does not need a ground. The coil under the flywheel is grounded to either the flywheel, to the crank, to the engine block, or directly to the engine block via a second wire, then to the battery negative via the cable.. The battery should also be grounded to the chassis so the lights, etcetera, can work. There is an off chance that the case of the regulator needs grounding.

BTW
This system cannot charge a dead battery. It only makes enough power to maintain it.