Sadly, firms abuse odd lot rules to give people terrible prices: https://academiccommons.columbia.edu/doi/10.7916/2y01-1s13/d...

In example 3, the NBBO for stock ABC is 495--500, but there is also an odd lot offer for 497 on exchange. If a Robinhood customer sends a market buy order, then Citadel is allowed to fill it for 499.999 even though it's better to send to the exchange. (And if they then pick up the odd lot themselves, it's easy arbitrage.)

By the way, while you're correct about some of your claims, odd lot executions definitely have to occur within the NBBO. (How could it be otherwise?) Otherwise, in the example above, Citadel would give an even worse price!