Exchange trading happens in round lots that are usually 100 shares.
This is pretty much just a legacy thing, but so many technical systems have this assumption built in that while odd-lot trading (trades not in the round lot size) has become a little more common on the exchanges, it’s still treated weirdly by the various systems involved.
But also, it’s better for you as a retail investor, to get them from a middleman, because they will generally give you a better price than the exchange. They will give you a better price because retail traders tend on average to be worse at trading than the overall market. You should take advantage of that, regardless of your actual ability level.
Odd lots don't contribute to the NBBO, and placing an order for an odd lot doesn't have to execute within the NBBO. (People can trade "past" you, I am pretty sure ISO's don't need to clear you, etc). (Note these are rules for market participants, not retail customers). So for a firm trying to argue they provide excellent price improvement and execution efficiency for their customers, they can't "just" send the orders to the lits.
And even if they could "just" do so, internal matching typically provides better price improvement on the NBBO than even the best execution you could get off the lits.
Edit: But yes TBC, you're correct that odd lot trades aren't unusual. But you're seeing trades there by actual market participants, not retail orders. They're not just trying to get those 2 shares, there's a broader strategy and they're aware of all the above nitty gritty.
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!
I mostly mean scenarios where your limit order might not be marketable, end up resting on the book, and then get traded "through". I'm speaking from the perspective of an actual direct market participant, where you're not using a market order but are trying to enter a position while adding liquidity/collecting a rebate. (Most exchanges reward participants who have some % of their trades as liquidity-added, with rebate tiers).
Round lots are excluded from the NBBO so that the NBBO can't be as easily influenced by quantities of shares that don't represent any material price signal. 1 share of practically anything but BRK class A represents ~nothing. Less than a round lot on a price level is basically no liquidity available at that level.
There are per ticker rules to allow odd lots on most US markets. AFAIK unless you're trading penny stocks, every stock out there is entitled for odd lots, and most trades are indeed odd lots, that has been the case for 10 years at least.
Even if there wasn't, I guess at least half the trading on stocks is through CFDs and not cash, so lots aren't even a thing for most investors.
This is pretty much just a legacy thing, but so many technical systems have this assumption built in that while odd-lot trading (trades not in the round lot size) has become a little more common on the exchanges, it’s still treated weirdly by the various systems involved.
But also, it’s better for you as a retail investor, to get them from a middleman, because they will generally give you a better price than the exchange. They will give you a better price because retail traders tend on average to be worse at trading than the overall market. You should take advantage of that, regardless of your actual ability level.