When that cheaper feed results in no one to fly the aircraft, yes, it could result in bankruptcy. As much as airline managers would like it to be true, airplanes don't fly themselves. Not for several decades at least.

If the problem was across the board in all airlines, then it would be an equal problem that works itself out. What happens to American if Delta finds a way to help their feed maintain solvency and continue to provide service cheaper than mainline?

It would result in a situation where one airline would be able to maintain both premium mainline service and also reasonable feed costs against another airline which cannot feed itself. Those flights to Heathrow, Paris and Shanghai don't fill themselves.