I would second the water pump flow rate, or a partially clogged radiator. Does the pulley mounted fan blades have a clutch Assembly attached, if so that could be possible cause by not getting it’s proper r.p.m.’s when it’s hot and idling.
The trouble with the fuel pump is that it is difficult to diagnose, because the machine itself could still be pumping but it’s only pumping fuel back in to the fuel tank, because the line to the fuel injector is blocked. My old car a '74 Datsun had this problem for years and no one diagnosed it until I went through about a dozen mechanics. All of them started with the vapour lock theory, because the car would only stall on really hot sunny days. Engine temp was high but fine. Getting a modern fuel pump fixed it for good.
Electrical things (not necessarily car-related) can fail when they get hot. Someone once suggested that the coil, being mounted on the engine, might be expanding in the heat and failing.
It’s not the water pump or any other part of the cooling system. There is no clutch on the fan.
The fuel pump is mounted in the right-rear wheel well, next to the fuel tank. There is zero possibility that engine heat is affecting it. And yes, it’s working. I always listen to it working before I turn the key to the Start position. 
It sounds like vapour lock is the issue.
Can you speculate/describe where the vapor lock issue may be occurring? Are you thinking somewhere in the fuel line or in the carb (not sucking enough air and not providing good fuel mixture)?
I don’t know. I’m guessing it’s the fuel line on top of the engine, or in the carbs. Since there are two carbs, I’m guessing… in the fuel line next to/into the carbs?
What type of carbs? Are they similar to the Strombergs, like in the Jensen Healy?
See the pic:
Another hunch.
When was the last time you serviced the carbs?:
- inspected replaced the needle
- checked diaphragm for cracks/leaks
- checked/refilled the oil reservoir
My thought is that the very hot air conditions may be too thin to provide sufficient venturi effect in the carb and you’re not sucking enough air/fuel mixture, hence the rough idle under these conditions. A cracked diaphragm might cause that. Or, something else that is interfering with the smooth operation of the needle/jet operation. Or, damned if I know. 
Vapor lock as I know it occurs in the inlet line to an engine-mounted fuel pump. The pump normally sucks the (liquid) gasoline in from the tank, but if that line gets hot enough the fuel inside turns to vapor, which the pump can’t draw in.
Ignition coils can indeed fail when hot. I’d try to test for spark when it won’t start. Also see if it will start and run (even if only for a second) on ether (starting fluid), if so it’s a fuel problem.
Cars in general continue to run even when overheated. While the less-than-ideal cooling system here may be a contributing factor, the root problem is almost certainly ignition- or fuel-related.
Worked on a Harley whos coil was the culprit. Bike ran fine, got hot, sputtered and died till it cooled off. This is an easy one to check if you havent already. Check primary and secondary coil resistance against spec when it’s hot enough to die. Doesn’t adress why the temp is rising though.
Oh, great. Now we’re back to the coil! :smack:
I don’t know how I’d test for a spark. The rubber hoods on the spark plug wires don’t really pull back. I have no wat to check for coil resistance, nor do I know the specifications. It never dies when it’s at the mechanics; only on hot days on a crowded freeway (which is not a good place to perform diagnostics). The only thing I can do to ‘check’ the coil is to put on a new coil and see if the same thing happens.
The temperature rise when idling in traffic on a hot day is normal for a car from 1966. It’s just they way they were back then.
If you have a heat gun you could try warming up the coil with the hood up and see if it fails. Next, try blasting the carbs with the heat. Just be mindful that a proper heat gun can get things melty pretty quickly. Alternatively, invert a can of air and let the propellant cool the coil and/or carbs to see if it goes away when you are experiencing the issue.
An easy way to check for ignition when it is hot & dies, is to carry an extra spark plug & install it into a spark plug wire pulled from any spark plug. Place the extra spark plug on metal in the engine bay or on the engine. Now crank the engine watching for spark from this plug.
I suspect the coil(S) myself. I have had this issue with an Austin Healey Bug Eyed Sprite.
Have you tried actually cooling off the fuel line when it appears to vapour lock? I had a '61 Falcon that would vapour lock on really hot days when I parked it and left it, then went back without enough time for the engine to cool. I learned to park it with the hood up when I wouldn’t be very long. But the times I forgot, I could get a wet rag, and cool off the fuel line, and then it would start. It was faster than waiting for the fuel line to cool off on its own.
BTW, for the car that never gets hot? your thermostat is probably stuck open.
I was a generator mech/maint in the army, and I worked on cars for a while after, and did little things on all my friends cars. Virtually every engine that never got hot (or car that never produced heat) had a stuck-open thermostat. I saw ONE with a bad heater core, which the previous owner had “fixed” by taking it out of the loop; I helped the guy get a junkyard heater core, install it, and put it back in the loop. It’s what happens when you buy a used car from a college student.
Johnny, are you sure it’s not just running too lean? Too much air, not enough fuel.
An odd thing that, but not enough fuel can cause an engine to run hot. Check your jets on the carbs. Perhaps get a bit larger jets.
What was the issue, Johhny LA? Did you work it out?