True, but the way I see it, the airlines and third party booking sites feel it's worthwhile to have their options appear on metasearch sites. The point I was trying to make is that there are examples for allowing "competitors" to use their API. Also there's a history in the travel industry of having shared booking channels in the form of the GDS systems (Sabre, Apollo, Shares, etc).
I suppose the difference is that there are a lot more competitors than in ridesharing, where Uber is clearly the dominant player. Since most people are aware of Uber, there is not much incentive for them to participate in third-party apps since if the choice is "use a third party app and always use the cheapest" or "just use Uber since the price is probably close enough not to matter", Uber prefers the latter.
And back to the airline industry, Southwest does the same thing. You may notice their fares don't show up in metasearch or third party booking sites. They have established a reputation as "the low fare airline" so it's not to their benefit to make it easy to see how their fares compare to others.