Earlier quoted context omitted.
The problem is that companies favor arbitration because arbitration tends to favor companies. If arbitration were truly neutral, and it were easy to access with low costs to the consumer, companies would stop using them. So even if companies don't pick the arbitrator, arbitrators still have an incentive to find in favor of companies. Additionally, small claims court already basically works like what you're describing…
Restructuring arbitration for neutrality and thus cannibalizing arbitration as an industry seems like a fine outcome to me. Feels a bit like the payday loan industry: I'm sure it does some people some good, but it's largely just predatory and throws people into a rigged game. We'd be better off with less of it.
My point is that it's hard to enforce neutrality, because there is always an implicit incentive for arbitrators to find in favor of big companies. Even if those companies aren't directly choosing a specific arbitrator, big companies in the aggregate are the ones primarily driving the business.