This shouldn't be the job of HR either.
What you describe is what civil courts are for. From societal perspective, it is their exact purpose: to dispense justice and prevent revenge. If they don't work, they need to be made to work, instead of relegating this function to random people with random training and random incentives following random policies that are ultimately designed to safeguard companies, not employees.
* the people are HR
* trained in sexual harassment prevention and response
* following a transparent policy approved by the board and a union vote (or something like it, for instance an employee rep on the board like they are asking for)
Maybe this isn't the best alternative. I'm just saying, it's not like the choice is between (a) the government works it out and (b) everything is random.
And employer is absolutely responsible for dispering judgement on inappropriate, but legal things that happen while at work.
If I go to a co-worker and start yelling slurs at them, I should expect the be fired, even if my behavior wasn't illegal. The same is true for inappropriate sexual actions.
Focusing specifically on the company's perspective, it's also a terrible option from a risk management perspective. It leaves them exposed to the possibility of a very expensive judgment. (Especially if it's a big well-known company that the courts might want to make an example of.) If they can resolve the dispute internally in a quick and satisfactory manner, then that is very much in the company's interest.
Which is exactly why it typically ends up being HR's job. And why it should officially be someone's job.
Framing it this way, I guess the company's internal culture being less dysfunctional because these sorts of problems are more likely to be addressed instead of being allowed to fester is just a happy side effect, but it bears mentioning, all the same.
Defining what makes a non-hostile workplace is hard, enforcing it on individuals is harder, so instead delegate that responsibility to each individual company and take action on the company if it does not do that.
In no way is it a perfect system (cue tons of excessively formalized training etc), but it is the best system I know of.
Allowing companies to remove the external force of "If you do not do this, you may face civil action" breaks the model though, so I completely understand why Googlers would like to remove forced arbitration. The risk of that is a contributing factor towards compliance.
I wouldn't shed a tear if forced arbitration were banned. I can't really see it as not being at least partially an attempt to wiggle out from under the rule of law.