What OT crap are you working on

That safety lever you tied down to make it run is attached to a kill switch that grounds out the power to the coil.
I'm suspecting a bad wire between that switch and the coil. That wire may ground out if it swings to just the right place. Or maybe the switch itself, or the linkage between the lever and the switch is messed up.
I'd start at the lever and inspect everything between it and where it cuts power to the coil.

 

Its a 71 Duster? Doesn't all the wiring pass through the bulkhead connector? Those are known to be corrosion collectors. Both gauges operate off ohm readings, and the wires for both gauges pass separately through the bulkhead connector (the bulkhead connector uses male and female spade terminals all bundled together into blocks of male and female terminal ends for every wire that passes through it). It wouldn't take much corrosion to cause a problem with either gauge wire as it passes through the bulkhead connector. I would start by inspecting and cleaning the wire connections at the bulkhead connection.

 

Only one of the wires passes through the firewall, the fuel gauge runs down the kickpanel and under the interior to the trunk. I'll have a look at the bulkhead connector for the temp gauge; cleaning things never hurts.

 

I knew I forgot to mention something...

The safety brake/lever gizmo has always been a problem with this mower since new. It works, but it's a junk design and material wise. It has been modified to work predictably.
Initially suspected that as a problem but it has been ruled out.
One last hurrah of the old coil was to snip the grounding wire and let it run to failure, and did so as expect ~5 mins.
New coil grounding wire was also kept disconnected and insulated to prevent unintended shutoff. Similar results, engine died after ~5mins.

My current question is how does the gas tank vent?
Only one fuel to carb outlet. And the gas cap is solid no venting.
Does air enter back up through the fuel line?

Maybe I'll do what all these flathead street rodders do, put some Amsoil 20W-50 in it

I am thinking of an oil change just to see if it's getting glittery.

 

Blowing the 69 apart

 

So I've been thinking, does the carb float bowl have a wire attached to it? I had an old Craftsman garden tractor that had a wire attached to the bottom of the float bowl. That wire was attached to the "anti-backfire valve" inside the float bowl. Its entire purpose in life was to make sure the motor didn't backfire if it was shut off at higher rpm. Whenever the motor was running, there was electric power to the "valve" and when the motor was shut off, the power ended. The electric power held an elongated plunger inside of the float bowl up to allow fuel to flow into the float bowl. When the power was shut off, the plunger dropped and stopped the fuel from entering the float bowl, the carb essentially ran out of gas (pretty quickly I might add). On my mower, the anti-backfire valve would just loose the electric power coming in and the motor would quit running. Sometimes I could mow all day without issue, sometimes it would die with every lap. In my case, shortening the plunger solved the problem. The plunger had what looked like two heads on it, I cut the top head off, and that problem went away. I have no idea where the source of that electrical power came from, and didn't really care.