Shielding that blocks high-energy gamma will not heat up significantly during the process of shielding because it absorbs or deflects the radiation, it doesn't convert it into thermal energy.
If it's absorbed it's converted into something. Conservation of energy and all.
Usually this means it winds up as heat.
The issue here is that if you are flying directly through the star's corona you are going to have to deal with superheated hydrogen that is millions of kelvin and is also in direct contact with your hull. That will melt most materials instantly, in much the same as if you were using a laser to burn through your hull, so presumably Elite ships must be specifically designed to handle very high temperatures while fuel scooping.
I still think you are confusing heat and temperature.
Temperature is only part of the equation. Temperature itself says exactly nothing about the total energy contained in something. You need to know the mass and specific heat capacity as well.
Air at normal pressure is a billion times as dense as coronal matter. The coronal matter in and of itself cannot appreciably heat your ship for the same reason you can't boil a lake by dropping one red hot coal into it.
The corona is so diffuse (~10^9 atoms per cubic cm) that even at millions of degrees the tiny mass in contact with your ship wouldn't have enough energy to be anywhere near the largest source of heating. Were I somehow completely shielded from the radiation of the star, I could hold my hand in coronal matter and not get burned because there aren't enough atoms to transfer heat to my skin faster than my body could move the heat away.
The heat radiators need to use some conductive method to get heat from the inside of your ship to the radiators on your ship's hull, i.e., likely some type of coolant system.
The radiators would have to be extremely hot to be able to cool the vessel purely by radiation, so there is clearly a heatpump of some sort at this stage.
The efficiency of the transfer of heat from your ship's heat radiators to the surrounding space will depend on the temperature gradient between those heat radiators and that space. When you're in deep space which is basically zero kelvin they will be at their most efficient, when you are in close proximity to a start at several thousand kelvin they will be less efficient and they really shouldn't work at all when flying through a star's corona because the corona temperatures are much hotter then the inside of your ship (i.e., if anything they would work in reverse and the corona's temperatures would actually heat up the insides of the ship).
You keep treating space like there is some sort of matter to conduct heat into.
The only practical way to cool an object in space is radiative cooling (
https://en.wikipedia.org/wiki/Radiative_transfer). Our ship's radiators are, perforce, vastly hotter than the inside of the ship, because radiative efficiency is so dependent on temperature and the radiators are so small.
ISS and Space shuttle use heatpumps and radiators as well:
https://www.nasa.gov/content/cooling-system-keeps-space-station-safe-productive
https://spaceflight.nasa.gov/shuttle/reference/shutref/orbiter/eclss/atcs.html
It doesn't make any sense at all to try to cool down the ship's hull because you have nowhere to transfer the heat. You can't cool the outside of the hull to a lower temperature then the surrounding space without heating up something else, i.e., the inside of the ship in the process.
Again, heat pumps. Ever used an air conditioner or a refrigerator? You can cool a space to far lower temperatures than the air you are using to cool the condensor. This is a form of heatpump.
Between the ship's radiators, which have to be glowing hot to work efficiently, there must be a heatpump to move heat from the relatively cooler interior and skin to the radiators.
Most likely the hull is simply a very good insulator and temperatures of thousands of degrees kelvin on the outside of the ship aren't conducted to the ship's interior. You could, however, achieve the reverse process to cool the insides of a ship by taking the high heat levels generated from the ship's reactor, conducting the heat through a coolant loop to the heat radiators and then radiating the waste heat into space. When you don't want to radiate that waste heat, i.e., in silent running mode, you simply close the heat radiator vents and become thermally insulated from the surrounding space and your ship disappears from sensors.
See above re. using some type of thermally-resistant compound to insulate spacecraft hulls. This is actually the same principle used with the heat shield system on the space shuttle that uses ceramic tiles. The ceramic material can withstand extreme temperatures generated during orbital reentry on the outside of the shuttle while keeping the inside of the ship from heating up because there is no way for the heat to be conducted through the ceramic material. The issue here is that if a tile breaks or becomes detached that is now a hole in your heat shield and the heat can now cause catastrophic damage to the inside of the ship. That's exactly what caused the destruction of the Columbia when they reentered orbit with damage to their ship's heat shield. Presumably a similar hull material exists in Elite to prevent ships from melting when scooping fuel and this would also be far more robust and damage-resistant then the technology we currently have available (otherwise any hull damage would make fuel scooping extremely risky).
No amount of insulation can replace actual heat removal/active heat rejection in the long term, the shuttle's thermal tiles included.
Since our ships can sit in close proximity to hot star indefinitely, the radiators are effectively cooling the vessel the entire time. If the hull/skin were not actively cooled, it would eventually overheat, and it would retain heat far longer than necessary.
Flying the ship with an external camera but no combat functionality isn't useful, the point is that we should be able to pilot the ship with the god-mode 3rd-person perspective and still have access to the HUD and weapons, just like the multicrew gunners.
That's a balance/flavor call on Frontier's part, just like countless other aspects of the game.