Because the trades are MATCHED. The bigger broker's have a better chance of finding another one of their own clients wishing to buy to match up against your sell order.
Since there are no MM's, there is no one to match up trades from all other brokers. Most brokers just do it in-house, but if you ask, and your position is large enough, they might contact another broker or two and let them know. But they are under no obligation to do so, and it would be on a broker-by-broker basis instead of electronically across the market where all brokers can see it.
The reason they don't guarantee a price in that case is because they will just match your sell against whatever price the handful of buyers are willing to pay. Which in a gray market stock, is usually not much.
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