To unclog a bathtub drain the right way, you'll need to take the drain plate off. That's the little round plate that has the lever on it for opening and closing the pop-up drain stopper. It takes a little effort, but it's not too hard to remove and behind that plate, you have access to a straight drain pipe you can send the drain snake down. Check out the video above from This Old House for instructions.
Plumbers in Australia have licensing requirements that differ from state to state but it is generally accepted a 4-year apprenticeship with a further minimum experience of 2 years (6 years total) and a further curricular requirement as a benchmark for licensing. Licensed plumbers are also expected to maintain minimum relevant training requirements to maintain their plumbing license