Webb isn't in an earth orbit though, it's at a Lagrange point that humans have never visited, and it's there because that's where it can get the best images. I don't think Starship can even get there at all without orbital refueling.
Sure, and it's also there because there was no plan to ever repair or service it. And that was because as it was being planned, launch was very expensive, so no satellite beyond LEO (where it could not be, for heating reasons) could reasonably be serviced.
That's not the main reason. Webb is viewing wavelengths where it has better viewing by being that distance away from the Earth. Putting Webb in similar Hubble orbit would diminish its abilities. If you want to build bigger visible wavelength platforms, then this might be a discussion to have with Starship capabilities.
WISE was another infrared telescope but in a low-ish Earth orbit (480km).
Putting it at L2 gives a more consistent thermal environment and means it doesn't have to repoint as often, but doesn't really make it any cooler. (After all, it's just barely past the Earth) At equilibrium the hot side of the Webb is around 300K. (80F)
Because it is an infrared telescope, and in earth orbit there are three very bright infrared sources that would mess up the instruments or heat up the telescope too much if they ever shone on the sensitive parts. (Sun, Moon and Earth.) When the telescope is placed in the Sun-Earth L2, all three sources are always very close to each other in the sky, and you can shield against all three with a single directional sunshield.
Because the JWST is not at the L2, but in a halo orbit around it, Earth in fact never covers the Sun at all from its point of view. They are just always fairly close to each other.
The coldhead of one of these sensors (MIRI) functions optimally at temperatures less than 7˚K (i.e. 7˚ above absolute 0). The other sensors run at temperatures under 40˚K.
For reference, the melting point of nitrogen is, 63˚K.
Any sunlight at all would overwhelm this sensor and severely reduce the operational lifetime (and results!) of the telescope.
At L2, the telescope has the sun and the Earth behind a layered sunshield. It's easier to maintain position there and to have both objects always in the sol-ward direction while looking outwards.
There are telescopes on Earth that cool their detectors (not optics, though) well below that temperature. The SPT-3G at the South Pole, for example, has detectors at 3K.
