I posted in your other thread about that Isuzu valve
LINK I used the aftermarket version. It doesn't have to be that particular sensor, any of the GM type with two nipples will work fine, too - that's what BTB uses when they 'restore' a later Air Cleaner. There's a pic in that thread. Those GM type valves are all over ebay as NOS. They all function essentially the same: when cold, vacuum is directed to thermo-flap thing to uptake pre-heated air, then when hot, bi-metal spring gradually opens to bleed filtered air into carb. In the FJ60, it's through the PCV system.
I think you're right, too, about so many trucks running around with that thermo-flap stuck in the closed position because every one of those HIC valves I've tested always fail to the default of vacuum to the flap, so it's always 'up'.
I only went with the Isuzu valve because it's made in Japan and was readily available on RockAuto, at the time, reboxed as various brands. I don't see it anymore

Looks like it has been discontinued.
Your cold-start issue seems more like one of the BVSVs is not working correctly (you still have all the smog equipment, yes?) but who knows. Could be at that temp (not something we normally see here

) without the air cleaner flap working, you're not getting proper fuel vaporization. It's also common for modern gasoline to 'capillary' out the venturi tubes overnight and pool on the bottom of the intake. Modern gas has a much-lower surface tension than gas from 40 years ago.
But all I can add is a LOT of the idle issues I was having went away when I fixed that HIC and thermo-flap on the air cleaner.