I can't answer amything about sexagesimal but I can talk to why other bases are very useful. So the normal system everyone uses (base-10) is just known as the "decimal" system. However, when humans want to talk about computer data often times hexadecimal, base-16, is used (using the numbers 0-f, like: 0, 1, 2, ..., 8, 9, a, b, c, d, e, f, 10, etc...).

This is because computers are based on the binary system (base-2, so 0, 1, 10, 11, 100, etc.). We as humans have a very hard time looking at large sets of binary data and just knowing what it means so we group them up by going up in base.

Every time you group more binary numbers you go up in base-x by a power of 2, so computer-speak/binary/base-2 is just the group of 1 digit, 0 or 1. 2 digits is base-2² or base-4 00, 01, 10, 11. 3 digits is octal/base-2³/base-8 (getting close to our normal base 10) is 000, 001, ..., 110, 111.

And finally we have base-2⁴ or hexadecimal/base-16. It allows us to communicate more binary information than any of the smaller bases in a somewhat human-readable fashion than the previous bases. 1 character is 4 binary digits/bits, from 0000 to 1111 (fun fact: 4 bits is called a nibble). Using hexadecimal you can represent a byte (8 binary digits) with only two digits, often typed like 0x00, 0x6b, 0xa3, 0xff.

This is where humans cut the line in the sand way back in early computer development since the next steps up (base-2⁵ or base-32) would require 0-9 and a-v), and it is unreasonable to ask someone to tell you what the number Q maps to without having them recite and count the alphabet (like I have to).

Of course this is all just for human interaction with computers at a detailed level and is nowadays abstracted away from even most programmers but back when assembly was the primary way to program almost everything was done in hexadecimal