360 degrees in a circle predates Plato by quite a lot (2000 years I think!). It comes from the Summarians more than 4000 years ago. They used a method of counting on fingers that goes up to 12 on one hand and 60 using both hands, so their numbering system was based on 60. 360 is 6 * 60 and also roughly how many days in a year.

Later societies inherited that from them along with 60 minutes in and hour.

But why wasn't the handcounting inherited too?

It sounds useful to be able to count up until 60 on two hands.