Yes, the nozzle is temperature regulated because the heater block has a heater cartridge (heat source) and the thermocouple(heat sensor) both connected to the mainbard and controlled by the firmware. The nozzle is THEN attached to that heater block.
So NO, the nozzle itself is not directly temp controlled, the heater block it screws into is.
That block and nozzle then must be attached b y something, and that is the thermal barrier tube.
Sorry,this pic below is a custom hot end prototype of a MK7ish design using copper for the heater block but is the same basic dimensions, The thermal barrier tube is an experimental one from Makergear they no longer sell. In actual usage, the tube gets cut down so the thinned section is right next to the heater block with minimal threads into the heater block. In theory, there would be a huge difference in temp from one side of the thinned section to the other (provided the top section is properly heasinked!!!).

That tube connects the hot end(the nozzle, heater block, heater core, thermocouple, and insulation to the rest of the extruder).

So one side of that metal tube is 230C since it screws halfway into the heater block and the nozzle meets and seals to it making a sealed path for the filament to hit the nozzle on the inside, be melted by the controlled heat and comes out the 0.4mm hole on the bottom side.
The point here is the thermal barrier tube has 5 very important jobs:
It connects the hot parts to the cold parts
It seals the filament path and nozzle in the hot zone so plastic doesn't leak out everywhere at the 2k+ PSI or more.
It attempts to limit the heat conduction from hot end upwards to the mounting block and motor (material choice and overall cross section, along with contact area at both ends determines how much heat is conducted)
Provides a smooth narrow guide path for the filament until it hits the melt zone. The filament is under compression load. The motor is pushing it down and 1.75mm filament can obviously bend sideways. If it does, it then rubs the walls and creates friction or resistance to actually reaching the melt zone.
So back to the assembled MK7 picture, the upper aluminum block is the one attached to the motor and must be cool and is colled by the heatsink and fan that bolts to the front of it, and even the motor body itself can be a cooling effect if the motor is cooler than the mounting block is being heated to by the hot end, via the thermal barrier.
And that's the real problem, We really do not care how warm the motor is, nor do we care about the block that attaches to the motor. What we want to cool is the filament path of the thermal barrier tube. It's being heated at one end to 230C. That heat wants to heat the entire thing to 230C just like the heater block and nozzle. The only way to cool that much heat is to conduct it to a larger chuck of aluminum (AKA the mounting block). If you don't have good conduction between the tube and the heat sink, it will be hot, that's just basic laws of physics. Heat wants to go somewhere, you must give it a path. Otherwise, it sits somwhere and that somewhere is the thermal barrier tube.