This is not at all how NBBO works. Effectively all retail brokers sell order flow and if they dont they still dont have any obligation to improve your price beyond NBBO.
All NBBO improvement requires is that if you send an order through the book you’ll get the best price on NBBO. Even when Robinhood got fined it wasn’t for that, it was for promising better than their competitors and then putting it in writing that they were happy being worse than their competitors.
Open to corrections
> they still dont have any obligation to improve your price beyond NBBO
But those that do price improvement use it as a marketing differentiator:
https://www.fidelity.com/learning-center/tools-demos/trading...
https://investor.vanguard.com/investing/online-trading/order...
https://www.schwab.com/execution-quality
https://www.tdameritrade.com/tools-and-platforms/order-execu...
https://www1.interactivebrokers.com/en/index.php?f=47202#bes...