A broker of mine does this, but the market maker polled price for selling is almost always higher. The quoted bid and ask prices (before the polling are just from the exchange order book. They are what’s left there waiting for someone to pick up. They are not necessarily the best prices available. On the other hand if I put in a large order, the prices polled are worse than the quoted bid and ask prices, the latter not being obtainable without breaking the order up into smaller chunks. And then the prices can move substantially over the course of the separate transactions.
To me, the bid and offer prices aren’t very useful. For valuation and performance purposes I’m interested in the price I actually paid and my idea of what price I would be prepared to sell the stock at currently (collapsing to what I actually sold at when I do sell). I use LTP rather than bid price as what I consider the best proxy for value when I have the choice in automated portfolio valuations. I never have to buy or sell at any quoted bid price. I can always choose to buy or sell at either my own price or not at all.