So Zillow is free to act as an intermediary on whatever terms are acceptable to the parties involved. On the other hand, Zillow's ability to act as an intermediary is greatly enhanced by access to MLS data. So if the NAR pulled Zillow's access to the MLS, a buyer would have a much harder time searching for houses to buy without a realtor.
Zillow gets their data generally from the various MLS around the country and aggregate data with public records
You are pretty much describing what a realtor does in the US. Whether it's a person whose beautiful smile adorns every local paper, a discount broker prmising to "SELL YOUR HOUSE FOR 2% FLAT FEE, GUARANTEED" or an online broker like Redfin, they're all bundled under realtor umbrella.
But to answer your question, Craigslist FSBO, yard signs offering FSBO or even Zillow "Make Me Move" self-generated listings are all doable, they just don't bring in as many leads.