If only it were that simple, but thankfully it is not. Otherwise the aero club mob really would take over all checking and training right across the board.
They certainly appear to have had considerable influence in CASA in drafting Part 61.
Your FE rating would need to specify the type e.g. Flight Examiner A-330. And probably somewhere it would say words to the effect 'simulator' or 'aircraft' to differentiate between one who does the type rating/IPC and one allowed to do line checks.
And because you are an FE on an airline type won't give you automatic privileges to instruct or examine on bug smashers.