ITA
Ah, yes, ITA.
A bit of history first.
In years gone by each airline had a separate fares system. But with improved communications some airlines started to sell their fares systems to other airlines with queries being fired from one airline to the fare quote system of a different airline.
Now, for airlines, fare quotes need to be competitive but they also need to be valid, in accordance with the fare rules. That's particularly true when airline A is providing fare quote services to other airlines. Note I said competitive, not cheapest, airlines may not act like it sometimes but they are commercial entities and want to make money and if cheapest isn't profitable for them maybe they don't want to expose that fare.
ITA started as almost an intellectual exercise for USA fares. They weren't selling their service, just exposing it on the web. Their USP was simple; cheap. If you looked at some of their results you found some weird routings and very long (and short) connections. And initially they were simply fares with no reservations checking so whether you could actually book their fares was an issue.
Word began to spread and the major US airlines and GDS began to take notice to the extent that some systems began to sign contracts with ITA to use their system for some fare quotes.
But the issue about whether some of the ITA quotes make sense except for total cheapskates still comes into play; different systems have slightly different "rules" about what they will and won't display so some ITA quotes get thrown away; it's usually related to the number of miles you have to fly to get that fare compared to the great circle difference. I mean, would you fly from London to Paris via Dusseldorf for a one day round trip which is an example ITA has just offered me with a return via Geneva with an overnight at Geneva?