It is very easy to just removed the fuel cap when this occurs to see if the engine will recover without waiting.
Or you can take a short piece (8" maybe) of insulated automotive 14 gauge wire and bend it roughly to a U shape.
Lay the wire into the open gas tank filler and hold the wire in place while you tighten the cap with the wire on top of the threads that hold the cap. This interference fit will permit you to operate the tractor without risking the cap falling off as could happen if you leave the cap loose as a test of tank venting.
If the engine runs past its usual time with this "vent wire" in place either find and unclog the cap vent or buy a new cap.
I have serious doubts that one of the magnetos is at fault.
loss of a magneto on one of these engines results in drastically reduced power but seldom kills the engine entirely.
Need to test that tank cap.

If the engine shuts off immediately, I'd suspect an electrical issue. If it sputters for a few seconds, based upon what you've already tried, I'd replace the fuel line to the tank.

I've had this issue a few times. It's not that the engine is hot, but that it takes a certain time for the fuel flow in a clogged hose to empty.

I found this: You are driving the machine and suddenly, it shuts off. So, whats the first thing you do? Remove the gas tank cap and look in. Hmmm! Plenty of gas. Start it up--it runs until a vacuum forms in the tank, and the engine stops. Can't remember how much gas was in the tank, so off comes the cap, you peer in, plenty of gas. Engine starts up, so off ya go to continue mowing, until--ooops, it stops running again! Ya think, maybe there isn't enough gas in the tank, so ya pour in a gallon. Starts right up--runs a while, then shuts off! RATS!!
And then, ya think about what ya have read about a plugged gas cap. So, ya remove the cap, stick a straightened paper clip thru the hole, and put the cap back on, and it runs and runs, and runs, until you shut it off to go eat lunch, and after lunch, it runs until you are done! (Of course--the lowly fuel filter takes a lot of blame for the problem--being that it never gets completly full of gas!) ;o) rustyj

Thanks for the suggestions. I wish I could see any sort of hint of a vent hole in my cap, but I can't. The cap makes more sense to me than the fuel line, only because of the length of time it runs. 30 minutes seems far too long to run a 16HP engine on the small amount of fuel that's in 3 feet of fuel line. Shouldn't I expect a whoosh of air when I open the gas cap, if that's the problem? I don't recall anything like that. I'm going to get an official plug for the float bowl and try the wire trick on the cap and see how that goes.

"***Shouldn't I expect a whoosh of air when I open the gas cap"***
No, it would not be a whoosh of air coming out........it would be a nearly silent rush of air INTO the tank when the seal of the cap is lost upon loosening the cap.
By the time the cap is all the way off, the VACUUM in the tank will have dissipated........so probably no sound at all unless you listen carefully.

I don't think it's the fuel solenoid. I replaced it with a metric bolt and it died several times last night during a 30 minute period. I'm almost certain that the real problem is somewhere in the wiring harness that affects the seat safety switch and the PTO. This is a 1995 Simplicity Broadmoor tractor. This harness is really loose. It doesn't snap in place; rather, it relies upon friction from 6 spade connectors and just presses in place beneath the dash. I also think the cap isn't the problem, but since I did open the tank briefly most times it died, I can't be certain yet. It'll be raining the next couple of days, so it'll probably be the weekend before I get my next shot at this.

Have you checked for spark immediately after it quits?
IF it seems to be a heat related issue, you probably have a bad coil.

You might check for spark with a cold engine first, so you have something to compare to.

If it's a bad coil that's weakened by heat, then I would expect that it would die very quickly again if I restart the engine immediately after it dies. It's actually running 5, 10, 15, or more minutes after restarting. And this engine has two magneto armatures. It runs too well for it to be starting on only one, and it seems improbable for both to be bad and weakening at the same moment. If only one was weakening from heat, I'd expect the engine to sputter as it ran on a single cylinder. For these reasons, I doubt it's a coil, but I may still try replacing them if all else fails.

Before mowing the front 3/4-acre this evening, I put a drop of Stabilant 22A contact enhancer on each contact in the wiring harness I mentioned in an earlier post. The bolt replacing the fuel solenoid was still in place. Gas cap screwed down tightly. Ran like a champ with no stalls. I'll try once more and if all's well, I'll put the fuel solenoid back on. I hope I've found my culprit once and for all. Hard to believe six drops of fluid could fix the problem, but I hope so. If that's the case, I can take some satisfaction in knowing that the carb is cleaner, valves are better adjusted, and the oil and oil filter have been replaced. There's only 665 hours on this Vanguard engine. It has a good reputation, it seems, so I expect it has lots of life left in it -- probably more than the Simplicity tractor it's powering.

If it was me I would do complete fuel system clean up. Said this before with that many hours and years on it your fuel system is dirty and about to fail.

I would Start at the tank. remove it, clean it out either by swabbing or pressure washing. rinse with gas or alky. Next I would replace all the fuel hoses, fuel filter, install shut off valve, replace the hoses after the fuel pump and the fuel pump vaccum hose.

If you don't want to clean the carb. bowl and spray the main jet with carb. cleaner add some fuel system cleaner when you get it filled back up with fresh gas and clean system up to the carb.

Also while I was in there I would also inspect the elect harness, switches, looking for loose hanging wires, chafed wires, rusty connections. plastic tie wrap any loose wiring that could get chafed or rubbed against moving parts. If you want it to last you need to do this very few years 6 is pressing it, but not bad depending on conditions you work your equipment in.

I would also make sure I check the air filter often older vanguard are fussy with the small air filter between the cylinders. Now cylinders reminds me of cooling fins. have you checked them lately? Might want to revisit the owners manual on pre-ventive maintenance make sure you got all the bases covered. do that and that tractor may last life time???? another thing if you going by the 50 hour oil changes you better be mowing in dust free invoriment?? if not then 50 hours between oil changes is way too long hence extreme conditions in the owner maintenance schedule??

Although I agree with you for the most part, the problem is that I would rather not lose a few toes or fingers, just to find out I was wrong about this. Until you are dead, you can never be sure these switches will not save you from some form of accident in the future.

I am an incredibly safe operator of equipment, as well, but I am also a human, susceptible to moments of lapses in my concentration.

Keep in mind, that as long as you are conscious of what you are doing, your motor will never switch off from a safety switch. Anytime my motor did that, I obviously was not totally concentrating on what I was doing.

So I have to disagree with you here. I like the safety switches and always blame myself when the motor shuts itself off, due to my mistake.

I am having the exact same issue in my Vanguard engine. I get 30 minutes perfect mowing and then a stall. After a 20 minute cool down I get 5 minutes run time. Over night I can mow another 30 minutes. Something is heating up and failing. I got 5 years of perfect running and now 6 months of half hour run time. Come on, Vanguard, what is going on? 28 hp fuel injected engine. Almost new.. Took the mower to the dealer and they wouldn't run it under load long enough to manifest the problem.