My first thought is, "Why didn't they eschew solar power entirely and 'simply' park it in the shade?"

Surely they considered that, and the wins would have been massive -- no sunshields needed. So I'm sure the downsides must have been massive as well. Not enough plutonium fuel, or perhaps it just wouldn't provide enough power over the life of the mission. And of course the radioactive fuel would generate its own heat as well.

Now I need to find out more about that decision....

edit: Another commenter mentioned, "The earth's shadow never reaches L2 anyhow - it's only penumbra at that distance since the angular size of the earth is smaller then the angular size of the Sun." If that's correct, then there was never truly an option of "parking it in the shade" anyway.

The dark side of the Earth is very bright in IR. Getting far away from the Earth and moon makes them dimmer. Also, going for L2 makes sure the Earth, moon, and Sun are all always in the same direction, so the sun shield will block all significant IR sources.

It can, it has batteries, but you don’t want to mess with thermals on that thing.