Bantamzen, the conflict at the Piccadilly throat, between Guide Bridge-Oxford Rd and terminating trains could have been (and partially was) resolved without the Ordsall Chord, by sending Liverpool TPEs to Victoria. What other problems was the Chord meant to solve which weren't airport-related ?
Yes, the priority given to serving the airport is not the only problem facing rail in the north, and yes, if decades of underinvestment had not happened we would not have this problem (though I suspect if the 70s plans had gone through miat airport passengers would be changing to some sort of metro service in downtown Manchester to reach the airport). However, as things are the airport is undoubtedly a subatantial spanner in the works.
One solution, perhaps as an alternative to extra Piccadilly platforms, would be the western airport link. The most obvious benefit of this would be that the Mid-Cheshire line service could be combined with stoppers to the airport from central Manchester. However, in addition the western link could be used for services to the airport from north Wales, Liverpool, Blackpool and Glasgow / Edinburgh. The north Wales and Liverpool trains could continue via the Styal line to terminate at the low numbered Piccadilly platforms. The Blackpool and Scotland trains (which would also connect Preston, Wigan and Warrington Bank Quay to the airport) could use the airport line as a loop, with trains which reach the airport via the western link heading back north via the Styal line and platforms 13 & 14, and trains which call first at platforms 13 & 14 continuing north by the western link.
In addition to removing some of the pressure on platforms 13 & 14, and a variety of other incidental benefits (connecting other lines to the field-west-of-the-airport HS2 station, enabling direct airport services from places like Runcorn, Chester, and Northwich) this would allow us to observe how many people actually use airport-bound trains to reach the airport, as opposed to central Manchester.