Actually, the biggest problem with base 60 is, if one is not using base 10 notation in base 60, the large number of symbols you would have to maintain (would make pocket calculators a bit unwieldy). Actually, not even the Babylonians used 60 discrete symbols.

Many people have long argued in favour of duodecimal (and it is not difficult to scale duodecimal to sexagesimal).

You are ofcourse correct that duodecimal has 3 different factors in its base, on the other hand, it just about balances out when you look at the base number, and those adjacent either side of it (it is fairly easy to determine if something is divisible by 3 or 9 in a decimal system because 3 is a factor of 9, and 9 is one below 10 - the comparable numbers for the duodecimal system are 11 and 13, both of which are prime).

The other advantage often quoted for the duodecimal system is in packaging. We still mostly prefer to buy eggs by the dozen and half dozen. We can now buy eggs by the 10, but we do not package eggs by 5. In many other contexts, it is often very convenient to package things in an array of 4 x 3, where it is far less convenient to package things in the more elongated arrays of 2 x 5. And, ofcourse, we still use duodecimal in many aspects of measuring time (duodecimal is a factor of the sexagesimal system we use, but is also a factor in the base 24 we measure hours in, and we still have 12 months in a year, albeit they are not equal length months).

Ofcourse, as you point out, there is increasingly a practical advantage in using hexadecimal as a base. Binary is very unwieldy (which is why octal and hexadecimal is much preferred as an appropriate balance between using a limited number of symbols while maintaining a more manageable length of number for most common uses).