Maybe because 360 is already divisible by 6, so 720 is not much of an upgrade over 5!. 7! On the other hand adds another prime factor to it.

7! 5040 has the less than useful property of being quite large for interacting with human scales.

5! 120 however lacks fine precision required at human scale. Haven't done the math but it's probably something like using 3.1 as the analog of Pi.

360 seems like it might have been chosen based on a mix of precision and practicality. Many small prime factors ( 2 2 2 3 3 5 ). Also an extra prior prime factor for every added prime. 75600 too big, and 12 what analog clock faces use as their primary number.

360 is base 60. (6*60)

Like minutes and seconds.

The 12 hours in a day and the 12 months are also 60 / 5.

This all connects to ancient Mesopotamia somehow.

> 12 hours in a day

I guess, for a sufficiently large value of 12.

Yes, the original day was 12 hours. (Hence the legacy of "AM" and "PM" that some countries still use.)