It sounds more like a matter of minor leagues versus major leagues.
Not really, because we don't really have that concept here in school sports.
The GPS is the association of the 'elite' schools of New South Wales - those schools that are widely considered to be the best and the most prestigious. They compete together in sport, but it's not primarily a sporting body.
St Edmund's has functionally been denied access to the best sporting competition it is currently eligible for because it's too good for the competition. St Edmund's is lucky enough that its reputation across the board is good enough it can make a serious plea to the GPS for a home, but that is likely to be a lot more than just a sporting change.