I've got a few filters that I picked up from Decatur. I think I'm out of the 'crudening,' but it never hurts.
A brief glance at the carb fan under the hood: The thermal switch under the intake/attached to the EGR piping has no wires running to it. That's for sure an issue. No sensor, no fan. I'll need to check the system out. (Note: diagram is for an FJ40.)

Hot restarts haven't been an issue so far, I'll chock that up to a dash of luck.
The issue arises when I'm driving.
Oddly enough, The rough idle/part-throttle operation issues subside after only a couple minutes with the engine off. i.e. Fight to get it in the garage, Chat with wife about it. Hop in the truck and it fires up and returns to idle RPM without choke.
Fuel cut solenoid:
Wiring at the carb is free of chews, and has been tested. She be clickin' when given 12V. Wiring from carb to engine bay/computer have not been tested. I should be seeing 12V at the female connection when keyed on, yes?. Given the smattering of wires that are just floating in space, I'd say it's a good place to start.
A few thoughts:
As temp increases, resistance also increases (yes?), that could lead to some intermitant fuel delivery issues.
An easy check would be spotting the fuel level in the bowl when the issue arises. If the solenoid is closing prematurely, fuel is cut off to the bowl showing a lower fuel level?
When the ignition is shut off, the engine follows suit immediatly.
Swapping that solenoid for a known good unit would probably be a good test as well.
Valve lash was set back before we ran into "The Troubles." I've probably put less than 400 miles on it since then. .008" and .014" for intake and exhaust respectively.