Here's the most systematic discussion on this I've seen for stock only: Aug 2022, SSRN-id4189239.pdf
From the conclusion:
It's pretty confusing but my understanding is that an order from say, robin hood and tda could be routed to the same market maker at the same time but because of the relative "market power" (vague), TDA clients get a better fill. Surprisingly PFOF amounts and from whom are not predictive of execution quality.
Apparently TDA has the lowest transaction costs among the brokers considered and very surprisingly to me, IB has the highest.