Ok to start from the top.
Your engine has a magneto ignition which is self energizing.
The key switch grounds out the coil to turn the engine off and goes open circuit to allow the engine to start & run
A relay is just a switch that is controlled by another switch ( or series of switches )
I do not have a circuit diagram for your mower.
However all mowers have 3 circuits
1) cranking circuit
2) ignition circuit
3) charging circuit
When you split them into these 3 fixing it becomes a lot lot easier.
The cranking circuit will go from the battery ( generally a take off from the battery cable to the starting solenoid to save wire ) througn the key switch, through the PTO switch throuhg the brake switch and if you ae a mechanical gearbox, through the Neutral switch then to the trigger wire on the starting solenoid.
This is a daisy chain so if any one of these switches remains open the mower will not crank.
The ignition circuit
This is a series of parallel circuits that will ground the magneto and kill the engine
usually
PTO + seat switch ( So you can not chop your own hand of foot off by clearing the deck chute with the blades spinning )
Seat Switch + Brake switch ( so you can not run over yourself if you fall off )
During cranking the set switch is out of the circuit but once you let the key return to run, it is in circuit so if the seeat switch thinks you are not there and the brake seat thinks the brake is not on then the magneto gets grounded & the mower turns off.
If the carburettor has a fuel shut off solenoid then you can remove the kill wire from the magneto and use the furl shut off to kill the engine.
However this is not a safe mannar to run the mower in but it might get you out of trouble.
On lesser mowers the seat switch & brake switch have 2 sets of contacts so the start & crank can be separate circuits for the brake and the seat can have 2 separate circuits, one to the PTO & the other to the parking brake.
On this model these switches are single pole switches and the cleaver stuff is done in the very expensive relay housing.
Yours sounds like it has seen better days'\
Usually they are just a few individual relays and some diodes and can be fixed if you can follow smoke trails.
Otherways it is new relay box or rewire the mower.
Both of which are worth doing as this is a very good mower.
Wat wat better than anything you will pay less than $ 5,000 for now days.
The caveat will be condition of the rest of the mower in particular how much rust there is.
The mower is in great shape as are the accessories (mulch kit, bagger). The only reason I am spending this much time on travel and diags is because the mower is a champ when it works. I wish I had pics of the mower to share, I have pics of every other damn thing lol.
Regarding the cranking circuit: since my motor does in fact crank over, is it safe to assume that all the safety switches are in fact properly closed? Could they be damaged-closed?
I didnt think the starter would turn over if the PTO switch was open (engaged) so I assume because the starter engages that the PTO was properly disengaged. The mower has a seat switch which worked last I knew; again it would not engage the starter if that seat switch was broken, so I assume that works too for the same reason as the PTO. Same with the brake switch. So because the starter cranks, I assume the PTO, seat, key, and brake switches are all closed. One might be broken, but it broke closed instead of open (if that's possible).
I do not know if it has a fuel cutoff solenoid. It has a manual fuel cutoff lever on the carb.
I bit the bullet and ordered a new combination relay for $130 as well as a new rectifier and key switch. I plan on immediately replacing the combo relay and rectifier, but since the key switch seems to be functioning I will hold off on that.
I will also check the fuse first, because like a dummy I forgot to look last time. I also picked up a new plug.
I appreciate all your help!