Correct, but their respective mission profiles are very different.
To my recollection, right now, the Air Force is in charge of maintaining the G2G nukes (in concert with the DOE) and the bomber squadrons that use A2G nukes. Yes, they also have fighters and whatnot, but that doesn't seem to be their primary mission anymore. Naval aviators seem to have taken the top shelf in that regard. Air Force also used to control space-based operations, but that mission has been uniquely reassigned to the new Space Force (with some residual Air Force involvement and coordination). Army and Marines do use aircraft, but largely in support of ground invasion, occupation & expedition ops. Coast Guard, of course, uses their aircraft exclusively for maritime rescue, mostly non-combative, provided they're not fighting off some cartel dinghies sailing across the Gulf of Mexico.