My local pizza place is vertically integrating marketing pizza, making pizza dough, making pizza, selling pizza, and delivering pizza.

It’s not the vertical integration that matters but the amount of market power a company has. My pizza place: almost none. Amazon: significantly more than none.

Standard Oil was vertically integrating around oil -- what is Amazon vertically integrating?

The entire sales process from searching & reviews to delivery. There are allegations that Amazon uses their data to identify the most profitable areas to set up their own competing brands and may even force sellers to disclose information which is helpful in this regard. If true, it seems pretty clear that a sensible antitrust rule would be to split those functions into separate businesses with strict rules about what non-public data they're allowed to exchange so the Amazon Basics guys don't have any information about what's selling where which you couldn't get.

Basics; AWS; Prime Video; Advertising.

You don't get to deny vertical integration just due to them being involved in everything.

Selling things. Almost all the things.

Selling things is selling things. That isn't vertically integrating. A good example would be selling and shipping now that Amazon is getting into delivery as well.

Amazon ships more packages through its DSPs than anyone. But I don’t see them doing delivery as anti-competitive. It is highly integrated and streamlined however.

They put a absolutely massive amount of resources on solving delivery. And they probably do it better than anyone. You couldn’t split that out without totally breaking it.