These things are not based upon logic, so much as history. Even to very recent times, the English used 12 pence to a shilling, 20 shillings to a pound. This complicates arithmetic, of course, but bookkeepers easily mastered it.
A reason for 60, like 12, is that it is easily divisible. So that in early times, it was used to facilitate making change and dividing up products. (Farmers, even in this country, favor the 12 makes a dozen eggs.) The Sumerians, and then the Babylonians, used the 60 system, but for hours it was 12 in the day and 12 at night. (This is not exactly the same as the 24 hour system, since some days are longer than others.) Where the Sumerians got the idea of 12 hours of daylight, is really hard to say.
I once hear a joke, "How many fingers do the flying saucer people have counting both hands?"
Answer: 10, we just don't know what base they use.
However, that idea, the universality of 10, hardly explains the 12 and 60 system.