It seems that given the options of 1) staying awake and active as normal during the night and 2) using the night period to run various maintenance task but without awareness the second option apparently turned out to be the winner as far as evolution is concerned.

What's interesting to me is how many animals beyond humans do it - so it's not purely tied to the human cognitive functions like higher order thinking etc.

I think part of the answer is that our world is just very conducive to hidey-holes. There isn't that much surface area you need to cover in a three-dimensional space to get protection from the elements and other predation, and there's an awful lot of stuff out there (trees, dirt, etc) you could comfortably use to do it.

3) Some animals can sleep with only half their brain at a time.

https://en.wikipedia.org/wiki/Unihemispheric_slow-wave_sleep

Plenty of animals are nocturnal, diurnal, or sleep in non-24h windows. Evolution does whatever works.