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.
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.
https://www.quora.com/Is-the-L2-point-of-Earth-in-a-permanen...
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)