So all this energy is coming up through thousands of kilometers of convection flows and suddenly, in some magic place, it all suddenly "cools off' to 4000K somewhere right before it reaches the umbra? Where? How? Where is all that extra energy going?
Asking how "all this
energy" suddenly "cools off" is very poor wording from a physical perspective. The better-phrased question would be "why is the matter in the umbra cooler than the matter in the surrounding, brighter areas."
As previous posts have explained, the umbra is cut off from the convection cells that dominate the surrounding areas. Thus, as the umbra radiates heat, it simply cools off because the matter in the umbra is not cycling back down to the deeper, hotter regions.
Of course there's still radiative and conductive heat transfer from below and from the surrounding areas. And, of course, the umbra itself is still radiating. But since the conductive and radiative coupling to the rest of the sun transfers quite a bit less energy than convection, the umbra's equilbrium temperature (where its radiation is equal to the energy it's receiving from the rest of the sun) is quite a bit lower than the equilbrium temperature of the non-sunspot areas.
If you think that it's simply impossible for convection to be cut off like that, explain why. If you think that cutting off convection couldn't cause the umbra's equilibrium temperature to be that much lower, I think you'll have to give us something quantitative rather than just asserting that this mechanism can't possibly give the observed result.
As for "it all suddenly cools off" - why do you think it's sudden?
If if the "mass flow" part stopped somehow (haven't a clue why), the radiation of heat from opaque area to opaque area would transfer the heat anyway from the below the sunspot.
Yes, it does. It just doesn't transfer heat as rapidly as convection does, so the sunspot's equilibrium temperature is lower.
ETA - I'll openly admit this is not one of my areas of expertise. Tim, Zig, Sol, RC, etc - I'm confident that you'll let me know if I got any of it wrong.