That sounds like perfect advice, and I understand it, but I don't hear any clicks at all.
With the machine in neutral, I turn the key on, and nothing. With the key on, I put in in 1st and back to neutral, and nothing.

Does anyone know what the Starter Circuit Relay looks like?

All I see are the regular Starter Relays, which look like this:


My Prairie doesn't have that 2nd relay. You can start it in gear, if you apply brake.
I wonder if I can just bypass that relay. But I need to find it first. The seat doesn't come off like normal machines, and I can't find anything on this one. Getting a little frustrated at this point! I need to find a service manual.

 

UPDATE: Although the battery tested good on the charger, I decided to double check because sometimes batteries have voltage but no amperage.

I pulled the battery from another machine and hooked it up. When I turned the key on, I heard a distinct click. Good start! Turned the key and she started immediately. YES! But there is a problem. After a couple of minutes of idling smoothly, the lights started pulsing, and the machine shut off. After that, the solenoid was buzzing, and it wouldn't start again.

There is definitely a charging problem. I had this happen once with an outboard motor, when the stator grounded itself. This seems like the same problem. It sucked the life out of a known good battery pretty fast, within 10 minutes.
How much energy does the ignition system use? Will it drain a batter pretty quickly?

I need to figure out whether it's the regulator, the stator or something grounding out.

Well... at least it's not the starter circuit relay that I couldn't find!

Any suggestions on where to start?

 

Be sure all the terminals on those thick leads are clean and tight, including earths to frame and engine. The ignition system won't take a big lot of power, so shouldn't run the battery down that fast, pity you didn't have the voltmeter ready to check if it was rising or falling when running. A heavy discharge would blow the main fuse anyway. I have a Honda 500 in at the moment with the stator shorting to earth, it charges, but only when you rev the nuts off it, and does not seem to discharge the battery much at low revs.

 

Good advice. I'll be following wires and checking every terminal and connection for corrosion and looseness.

Curiously, the original battery read 13.8V when I first checked it. Now, after being on a charger overnight, the red light is flashing, "Bad." Seems like an internal short in the battery. It came with the machine, so no idea how old it is. Those kids were running it hard. I'm thinking the battery got an internal short first and it made the stator overheat. The 2nd battery is several years old, but it recharged with no issue and is fine.

I have a new AGM battery coming tomorrow. I'll check connections, put it in and see what my voltages are when running.

Thanks for your help and advice, Merryman. I appreciate you sticking with me.