Machine shops need bodies to do a few things (some of which OP mentioned) loading materials, removing materials, changing broken cutting bits, cleaning the machines (especially if you're cutting something like magnesium).
Machine shops also need bodies to build jigs to hold pieces to the machine table while the machines remove material (think 5 axis pieces, especially).
Despite all of this – there's an amazing amount of automation at play. Tool changes, coolant, temperature monitoring, the list goes on. I'm currently building some automation hardware and software so we can start to fill the gap of 'dead' time where machinists are pressing buttons, loading in blanks, and are bored out of their minds.
It's such an interesting space.