I think using your spark tester is the best approach.

Posting the Sears 917.xxxxxx number would allow one to look at a schematic (if available) for any other potential issues.
The tractor may have an Operator Presence Relay that could possibly be losing contact or some other similar obscure problem. A schematic would allow one to see if you even have an OPR.

***"rural property, has never died from being hot, only a restart issue, restarts when cooled"***

Before going any further with this, pull the sheet metal shrouding and check for mice activity (nest building, food caches).

Mice activity is often the cause of overheating. The critters can destroy an engine if they pack enough material into the duct work.
Even small amounts of material can block enough air flow to create problems.
And yes, use a spark tester to confirm the loss of spark if you don't find any duct work blockage.

Bill: Thanks for the reply! You walked me through some OPR issues last year on a similar mower and I'm pretty sure it's not a safety switch issue.
Mownie: Thanks! I had actually cleaned the mouse nests out a couple of weeks before on this mower but I guess they can rebuild pretty quickly LOL. I'll take a ratchet and sockets with me to pull the shroud to verify though.

Well, if I "walked me through some OPR issues last year on a similar mower"
I wouldn't have done it without the 917#, which I requested for this mower.
Your call-

You're absolutely right! I did furnish you the number and you were very helpful, explaining how the switches worked for the safeties etc. I can't get to the 917 number on this mower until I get to the property. Thanks

"Starting fluid will NOT make it fire"

"Am I right in thinking it can't be a fuel problem if the starter fluid won't help it fire?"

It could be that the carb float needle is not seating good, maybe has a speck of dirt under it and the engine is flooding when stopped. I would try shutting the gas off 10 seconds or so before stopping the engine and see if that helps.

Walt Conner