If vertical integration is always good, the ideal situation is a single producer for any kind of product and no suppliers, which doesn't seem to work too well either.
In markets where firms have market power, there is an explicit and well-studied phenomenon known as "double marginalization"[1][2], which is what the parent comment is describing. Vertical integration is one possible solution to this problem, although there are other approaches as well.
Double marginalization also occurs elsewhere in the air travel industry. Imagine I flew on American Airlines from Charlotte to London, and then continued on British Airways from London to Lyon. If both airlines set their prices independently, then they would both add a markup, and the combined price of the itinerary would be higher than the profit-maximizing price that a single airline would set. In this situation, antitrust authorities will often allow the airlines to form a joint venture and coordinate on price-setting. This reduces horizontal competition (because e.g. AA and BA are no longer competing with one another for passengers on JFK-LHR), but can provide consumer benefits if eliminating double marginalization allows airlines to set their prices lower on multi-airline itineraries.
[1] https://en.wikipedia.org/wiki/Double_marginalization
[2] https://mru.org/courses/development-economics/double-margina...
In a few cases, economic theories do a passable job of roughly mirroring real-world behavior, but that's the exception and not the rule.
When a principle is violated marginally, conclusions made based on the principle are themselves violated marginally. You can't write off chemistry because relativistic effects violate the conservation of mass any more than you can write off economics because no two soup cans are identical.