I don't view the Chase lounge as a competition to AA FL since it's over in D where, I'm guessing, they want to capture (in part) the UA crowd in the Chase ecosystem. Even for those AA fliers who have a Chase card that gives access to the lounge, it's a little out of the way unless someone has a lot of extra time and wants to get their steps in for the day....
It’s also not a competition because whatever you may think of AA, Chase’s flights are far less convenient and they are far stingier with upgrades. So if you’re flying AA and chose to access the Chase lounge rather than the FL, I don’t imagine AA is going to object.