Base sixty would be much worse for humans. There are too many digits to memorize, and the multiplication table is enormous. It does give you a factor of 5, but 5 is only so common because we use base ten.
Base twelve is better than thirty because you can represent 1/4 (0.3) and 3/4 (0.9) with a single digit.
We do, for physics. Very often c is treated as 1. The "meter" is really just a conveniently human-scaled unit, equivalent to the "degree" in trigonometry.