Why does H3 use a nonoverlapping set of hexagons? A square grid would make it even simpler and faster to calculate. I am perfectly happy to believe that a hex grid works better for some reason but what is that reason?

Maybe this comparison with S2 will explain:

https://h3geo.org/docs/comparisons/s2/

The main design goal was to make the distance between neighbours constant. With squares, you have 4 side neighbours and 4 corner neighbours. With hexagons, it's easier to interpolate paths and analyse distances.