Even if it were true (but it's not) it wouldn't solve the problem anyway as the RAAF base airspaces (tower and approach) would still need to be staffed. So while there may be some redundancy, it certainly wouldn't be the entire RAAF ATC workforce suddenly jumping over to ASA.
There would still be the same amount of airspace, the same number of towers and terminal areas, maybe just some reshuffling of airspace volumes to free up some staff. However, working on the Richmond model (where APP is provided by Syd Dep West IIRC) it wouldn't be an acceptable level of service for the military.
No offence to the SYD ATCOs, obviously they are limited by understaffing, but when it takes 5+ mins to process a "next" call when there is nothing visibile on radar in the way (because the SYD controller is also tied up providing a RIS/RAS and/or combined with DEP east etc and simply can't respond to RIC TWR) it gets very frustrating for the mil.
As was posted earlier - the RAAF is run by pilots, not ATCOs, so delays (= added fuel costs, missed TOTs etc) at the sharper end can not be tolerated - hence mil controllers and truck loads of mil airspace.
I would love to see one ATC service provider for Oz, but not if it's ASA.