totally understand the Gold Coast not representing the whole of Australia,
The Gold Coast doesn't even represent the Sunshine Coast and they're only a couple of hours apart. Not a fan at all of the Gold Coast! Northside the whole way here.
It's a little brisk in Qld at the moment, but overall this winter has been really mild. It's only been under 5 degrees out at the horse for a week or so. The further north you go in Qld, the more tropical it is.
The east coast is pretty good. To save time getting from Sydney to Brisbane, I'd recommend flying - it's an 8+hr drive.
Giving a week in Sydney and Blue Mountains, you could go to Brisbane for 5 days. Day trip to Lamington National park, nice bush walk through Springbrook, Mt Tambourine.
You could go up to Glass House mountains and climb one, over to Bribie Island, Stradbroke island, Noosa, Rainbow beach. Australia Zoo (Steve Irwin) is up that way and you can spend a nice day there.
Fly north from Brisbane to Cairns (it's about 20hrs to drive). You've got the Daintree, Port Douglas, the great barrier reef.
Or you could fly from Brisbane to Alice Springs (32hr drive) and see Uluru, the Olga's and Kings Canyon.
I'd pick a place as a base for a week and do day trips. Where you go depends on what your family enjoy doing.