They do, and AA does from DFW and CLT. DL does from markets that it owns (DTW and ATL) and is willing to deal with JFK. Maybe they have a corporate contract in NYC that helps it. Delta has also been in Germany for nearly 40 years and has worked at building a point of sale strength there to get some of the German point of Sale. AA has been trying for less time, and for the most part has failed to capture the German point of sale corporate market, for whatever the reason.
Not all airlines can be all things to all people, and you see that in the various decisions that each carrier has to make as they're figuring out how to most profitably allocate their resources.