"the problem is with degrees"

I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier.

So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10.

I think the point is to be able to teach geometry to kids without worrying about them getting confused by fractions and/or irrational numbers.