The more specific we are in our discussion the better. So let's start with brass tacks, and with Amazon.
Amazon owns a marketplace, and it also owns a warehouse and shipping service, known as fulfillment by Amazon. It also owns an advertising business.
Recent testimony in Congress revealed what sellers have long suspected: Amazon gives preferential treatment to sellers who use its warehouse and shipping service, Fulfillment by Amazon. (Sellers also suspect that Amazon gives preferential treatment to sellers who use its advertising service).
You search for "toothbrush" on Amazon? It will serve up toothbrush sellers who use FBA before those who don't.
It will likely also serve up sellers who use its advertising service before those who don't.
What that means is that sellers, who depend on high product placement in search results to survive, "choose" to use FBA in order to get the better placement, and "choose" to pay extra for paid advertising to improve their so-called organic search result status. As a result, Amazon takes on average 30% of every sale on Amazon, an enormous cut, and an enormous wealth transfer from small businesses to Amazon.
On the worker front, it means that Amazon is rapidly building warehouses, taking the business that was once performed by mid-sized manufacturers and retailers, and could be again. Once it dominates in the warehousing industry, it then uses its market power to engage in union busting, making it difficult for workers to organize, which in turn reduces wages and benefits, and makes it more likely that workers will be spied upon.
Because of the scale of the operation, it has become a kind of private government (something that sellers report they experience), with its own intellectual property regime, which it uses to build power and extract wealth from sellers.
So, yes, we should break it up, and require it to divest from the ancillary businesses. The federal government or big states could pass a law that a platform of a certain size cannot also be engaged in ancillary services, a single-line of business law, that reduces conflicts and wealth transfers.
We could, alternatively, sue it for violating tying laws, and work to strengthen those laws (both of which I support), but the reason breaking it up is important is that it removes the conflict, a far simpler tool than constantly hunting for evidence of tying, when the incentive structure is clear. Amazon's ability to cross-subsidize and engage in predatory behavior could be dealt with by a series of predatory pricing and inputs cases, but it is simpler, again, to acknowledge the structure of incentives and simply break it up.
Google can be broken up from Google Flights, Google Shopping, and YouTube, and free up the space from conflicts, which lead to wealth transfers. Facebook from Messenger, John Deere from financing, etc.
Horizontal breakups are another matter, and there I am more skeptical, and prefer common carriage regimes, but the status quo must change.