Ron, the problems may be related, let's look at it.
Step One:
According to what I'm seeing on the schematic, the tractor indicator light #1 (in the schematic should come on to tell you to "Depress Left Pedal," that is your brake. When you depress your brake pedal, you make your brake switch which passes power from the "Start" terminal on the key to the PTO switch at terminal "C" (red wire) If the PTO switch is in the "OFF" position, then the power is passed to the starter solenoid from terminal "D" (also red wire) on the PTO switch.
If your starter solenoid is not pulling in, there is a problem, in reverse order, with the grounding of the solenoid, with the PTO switch, with the brake switch, or the the key. Check all the terminals, clean the grounds, usual drill.
Step Two:
When the PTO switch is in the "Start" position, then the Alternator is connected directly to the electro-magnetic coil on the PTO by means of the alternator at terminal "F" (red/white wire) on the PTO switch to terminal "A" (brown wire) on the PTO switch.
At the same time, the brown wire to the PTO clutch is also supposed to be seeing the same alternator current from the Normally Open contacts of the Reverse Relay. Notice that the alternator red/white wire splits just before the PTO switch and runs over to terminal 30 of the reverse relay. The same current coming in should leave the relay on the other end of the brown wire at terminal "87" of the Reverse Relay.
Step Three (the key point)
The alternate path of the alternator current is made possible by the PTO switch in conjunction with the Normally Open Seat Switch. Again, in the "Start" position, the alternator current at the PTO switch terminal "F" can flow to the clutch from terminal "A" at the PTO switch, but it can also, if someone is sitting in the seat, and the reverse relay is energized, also flow from terminal "87" of the reverse relay out to the clutch by the alternate path.
When you release the PTO switch from the "Start" position, the alternate path for the current becomes the only path for the current. So both your Normally Open seat switch (the one with the purple and black wires) must be made (show continuity) and the reverse relay must be pulled in.
When you are not in reverse, the reverse switch provides a path to ground for the alternator current not being consumed by the PTO clutch and coming back through the PTO switch on the purple wire at terminal "B." When the transmission is put into reverse, the reverse switch is made, the connection to ground is opened, the "In Reverse" indicator light comes on, and the reverse relay drops out. When the reverse relay drops out, there is no longer any path for the alternator current to get to the PTO clutch, so the clutch drops out and the blades stop turning.
Note: The purpose of the PTO switch's "Start" position is to pull in the Reverse Relay. In electrical terms, the "Start" position of the PTO switch energizes the Reverse Relay, and the Reverse Relay is latched when the PTO switch is in the "Run" position: the output is fed back to the input to keep the reverse relay pulled in.
All of that being said, it sounds like there may be a problem with the either the normally open seat switch, the reverse switch, or the reverse relay itself. Make sure the coil on the reverse relay is good (there should be a low ohm reading between terminals "85" and "86" of the reverse relay). Make sure your seat switch is being made, there should be continuity across the switch when someone is sitting in the seat.
I'm not going to tell you to jump anything out, but sometimes for diagnostic purposes, it is a good idea to make the circuit as simple as possible in order to isolate the failure.
Note: It is always possible the PTO switch is bad, but many switches have been replaced before the owner discovered the problem lay elsewhere. PTO switches aren't cheap.
I also recommend Roland Bedell's excellent portrayals of the various PTO switches used in classic Cub Cadets. You can find his colored drawings in the Wiring Diagrams section off the main forum.
There is also wiring schematics for most all Cub Cadets in the Wiring Diagrams section from whence I pulled the schematic below. They are much more legible than the snap shot I've provided.
Good luck.