In irrops an airline will take over a ticket but generally no, you have to stick with your ticketing airline. Travel agents can't take over an airline issued ticket either. I can imagine an enterprising age t could make a fair amount from all these QF issued tickets if that were possible, haha! Even in irrops I suspect the operating airline would make only the necessary change then send it back to the issuing airline. I haven't personally experienced any disruptions beyond the typical schedule changes on one of these so can't say 100%.

In your case, AA is much easier to deal with on these tickets than CX, no reason to try getting someone else to take it over.