Typically when you buy/sell a stock, it's not moving anywhere. Robinhood would try to sell the stock to someone on Robinhood buying the stock first. The big banks and brokerages have enough people buying/selling within its platform that it has enough supply to trade within itself.

For a small brokerage, they would have to work with other brokerages to help them trade, as they wouldn't have enough shares they already hold to trade them within their users. So Robinhood asks Apex for more shares, and Apex has to ask HSBC/Fidelity/JPMC/Goldman if Apex doesn't have any. At some point, all the other institutions also don't want to give up their shares because they need to satisfy demand from their own users. Trading with other firms tends to be really slow too.

It's a short squeeze too, by definition there's not enough stocks available for everyone who wants to buy. So some firms ran out and couldn't get more.

You left out the exchange. Why can't the small brokerages go directly to the exchange?

Defeats the purpose of using a broker/clearing house, there are products that let you trade on the exchange directly. RH probably figured there's no point in integrating all the way to the end if no one would normally use it.

Same with doing your own clearing, there's complexity involved with clearing and maybe you don't want to implement it all yourself when someone does it better for cheaper.

It's like asking why we don't all program in assembly all the time, or even have the option to mix assembly into our high level code. It's just nice having layers of abstraction that handle the annoying stuff for you, even if there's a minor cost. Why have complexity you don't expect to need?

You may want to look up Reg NMS and the various rules about internalizers.