Thanks for all the feedback thus far. To summarize, it seems that there are 5 possible culprits:
1. Seat Switch
2. Brake Pedal Switch
3. PTO Switch
4. 3 Relays (Start-Kill-Interlock)
5. Ignition Switch
My (amateurish) analysis thus far:
1. I neglected to inform you that I had bypassed the seat switch years ago by inserting a jumper wire. Yesterday, I removed the wire to see whether that had any effect. It did, the tractor would not start at all without it. I replaced the wire, and now the tractor will start again, although it continues to shut down. Thus, I don't think this switch is my problem.
2. The brake pedal switch seems to be working properly. The tractor cannot be started without depressing the pedal. Even if I keep it depressed, when I turn the key to start, and then release the key, the tractor stops running. Again, I don't think the brake switch is my problem.
3. As for the PTO switch, the tractor will not start unless it the PTO button is depressed. If I start the tractor and keep the key in the start position, then pull the PTO button, the snow thrower starts to run for a split second, but then the tractor shuts down again. This switch seems to be working properly, although I guess I can't be certain.
4. I haven't yet tried switching the relays to try to determine if one is bad, but I will do so momentarily.
5. The ignition switch was replaced a few years ago, but it could have gone bad in the interim.
I realize that problems like this sometimes need a "trial and error" approach. However, if I find that the relays seem to be ok, then I will replace the ignition switch. If then I still have the same problem, I will report back.