Great suggestion but unfortunately doomed.
ASFA tried to pull together an organisation like this about 4 years ago and in fact got it through the formative stages. It then withered on the vine as apathy and each and every secular component of aviation decided that either they alone represented aviation (AOPA by way of example but not alone) or that they just didn't want someone else representing their individual voice (such as QANTAS but also not alone).
It even got to the stage of meeting with the minister and the then head of CASA Bruce B. I was there and can still remember Bruce saying it would be great to see it succeed but his closing words was good luck as he didn't think it would for all the reasons above.
In the past there was the GAA which was as close as a GA organisation got I would think.
If you want to follow it through talk to someone from ASFA that was there then. No use relearning mistakes already learnt.