Longer wavelengths require cooler temperatures to detect efficiently. The MIRI detects all the way out to 25 μm and so needs to be cooled to 7 K.
Webb can passively cool down to 40 K with its sun-shield, which is very good because at 40 K the black-body radiation is 3000 times weaker than at 300 K. A cryocooler is necessary to cool MIRI further.
A cryocooler is a heat pump, using energy to pump heat from a cold to hot place. The greater the temperature ratio the harder it is to both pump the heat and keep heat from conducting backwards.
The hot end is at 300 K in terms of electronics and pumps, even though passive cooling can reach much lower temperatures.
Does the helium in the compressor approach 300 K? The working fluid hot end, not the hardware temperature, should be kept cold to reduce the load on the pumps.
If so, why? Is it too difficult to passively cool the working fluid to ~100 K or so while keeping the "noisy" pumps far away from the instruments?