I don't think the choice of base is really dependent on finger encoding efficiency. Going 0-5 on a hand has the advantage that we only really care about the "last" finger.

Anyway, if we really want to go crazy, the knuckle and first joint on (my, at least) fingers can be controlled somewhat independently, so I think we can fit in 4 states per finger:

  |   _
  |  |  __ _
            |

Try using that to explain arithmetic to a child.

In a base-10 system the explanation is as simple as holding up both hands and then holding up the number of fingers for each hand you want them to add. The kid can count and intuitively learn how to add up two numbers. It’s really simple and really efficient.

You might be able to. Personally I can't raise my ring finger without my little/pinkie finger also being raised.

Also, some numbers in finger binary are liable to get you punched if shown the wrong way round.

A few civilizations have used base 60

https://en.wikipedia.org/wiki/Sexagesimal