IANAL, but why not just have a special labor court with more judges to be the nonbiased arbitrators? Or use retired judges to be like “independent contractor” arbitrators as a special division of the courts?
(I’m also not sure what the requirements to be an arbitrator are so please correct me).
This is what we have in Germany. It's called "Arbeitsgericht" ("labor court" basically) and as an employee you can go there and sue your employer.
For employees this is free in the first instance and you will have a neutral judge to rule about the situation. I think you don't even need a lawyer as an employee but just "file a case" with the court, but I'm not totally sure about that.
Anyway, it's a very important part of the system in Germany, those courts keep a lot of cases with work-related topics away from the "normal" courts while still providing a neutral platform for both parties.
In England and Wales there are Employment Tribunals. These are run by a government appointed judges in an informal manner that should let people represent themselves. They include a conciliation service.
There are time limits on when a claim can be made and - if monetary compensation is ordered - how much can be paid out [1][2].
In 2012 the government imposed a £1200 (from memory) fee for taking a case to tribunal. So a low wage employee without union representation who was harassed out of a job or unfairly dismissed would have to: File a complaint while looking for a new job. Pay the fees during or after a period of involuntary unemployment. Prepare their case while working their new job. Arrange time off for themselves (and witnesses) and pay for transport to the hearing(s). Make their case. Possibly receive compensation totalling a few thousand pounds.
Claims dropped a lot and it was five years before the Supreme Court told the government to knock it off.
It's rough (and precarious) at the bottom. There have also been large cut backs in legal aid for dealing with the "proper" courts (and not just for civil cases[3]).
[1] The tribunal can also order an unfair employer to give the employee their job back or change their employment conditions.
[2] I think there are differences for cases involving some forms of discrimination.
(I’m also not sure what the requirements to be an arbitrator are so please correct me).