The airlines operations, shit as they are in the US, agreed aren't that "profitable" on the whole, with some exceptions like Delta and Alaska...but you're leaving out the fact that their immensely profitable "division" let's call it, where does that money go? Shareholder buybacks, stock grants--and certainly not making the flight experience suck less which might actually induce demand. Airlines are expensive and cyclical to run, but when they are profitable it's not used prudently because they know they can just get a government bailout so why not pay off their rich friends in the meantime and juice stock price?
Your line of thinking is like saying all products in a business are equally profitable. Grocery stores have higher margin products and lower but it doesn't mean they don't need them all to compete in the marketplace--they can't just have a loyalty program without the airline. If they could they would have long ago done it (and to a degree they have selling huge amounts of forward miles to banks) but at that point there would be no argument left that they are operating independently of government backing.
They were regulated utilities, then the champion of "free markets" Reagan, deregulated them leading to the front capitalism we have today; private gains and public-vscled bailout losses. Funny the government never gets called into the fat years to take dividend checks from airlines...