Arbitration means that you don't get to sue in real court, but instead you are forced to go through "arbitration", which is a private company hired by your employer that's supposed to make fair decisions in disputes between the party that brings them continued business, and the random person that will never talk to them if they weren't forced to.

This is not correct. You are not obligated to agree to the arbitrator your company selects. More on this:

http://help.legalnature.com/arbitration-agreement-faqs/how-a...

Isn’t sexual assault / harassment a criminal offence though? Surely the state should be prosecuting the moment a law is broken?

Or does the US justice system work differently?

There are plenty of fireable offenses of sexual harrasment type that don’t meet the relevant criminal standard of sexual assault. Sexual harrasment is not a crime, though often it is very close to the crime of sexual assault (e.g. unwanted touching is a crime).

The Federal Arbitration act was specifically setup to provide a lower cost avenue to settle disputes as it is widely agreed that the current court system is too expensive for the average person.

There is valid criticism that it prevents class action suits, but aside from that arbitration is better for everyone involved. Most people won't even bother to litigate a lawsuit due to the costs exceeding the relief sought, but arbitration is far more approachable.

This is where organized work forces (most commonly labor unions) show their worth. It's common practice that if you are represented by a labor union, the arbitrator (private judge) is mutually agreed upon by the union and corporation, which helps mitigate massive bias in either direction. Most US companies that mandate arbitration, the employees have zero voice in selecting the arbitrator.