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.