An MLS is nothing but an local organization. There are zero laws requiring a property to be posted on an MLS. I know of zero states with any regulations like what you propose. There's also confusion around the term "Realtor" because "Realtor" isn't actually a thing. It's a brand, a "real estate agent" on the other hand is an actual job.
The real estate industry is *truly* and exceptionally fucked up. The confusion around MLS, "Realtor," and probably loads of other things I'm forgetting are intentional.
There are similar private requirements for the cartel known as NAR. What this means in practice is that pocket listings actually make to to the MLS for one day before apparently being sold "in 1 day".
Zillow as property owner is not part of NAR. Zillow as broker is also not part of NAR, although if they were, they'd do exactly as individuals using pocket listings do. Arrange a sale privately with an institution, place the listing on MLS, then sell it "on the open market" in 1 day.
Today they essentially function as data brokers and unofficial labor organizers, collecting membership dues from agents in exchange for ensuring that the agent's name is on their listing wherever it's shown online and (along with NAR) pushing agents to keep "standard" commissions on their listings (5-6%).
How do you know without getting bids from the open market ?
You can see the transaction history here:
https://www.zillow.com/homedetails/125-Mays-Blf-Fayetteville...
Add in whatever fees go along with the sale, and they're out a fair bit more than that.
Opendoor still holds it, Zillow is already marking down a similar one by 14%.