This thread has some more info:
Your employer is either clueless or trying to increase its control over you – nothing unusual either way. Of course, if you actually want an ARC before your WHV expires, go ahead and get the work permit. With an ARC you will be subject to NHI, regardless of your private insurance.
If it’s a buxiban job, whether you qualify or not depends on what the Workforce Development Agency thinks of your teaching certificate (since you don’t have a degree). In your case (as an Aussie), the advantage to having a normal work permit is that you can stay with the same employer for more than six months. The disadvantage is that you will lose your right to take other jobs, unless you get additional work permits, which will be subject to restrictions: 6h/week for each additional buxiban, and for non-buxiban jobs you generally won’t qualify for a work permit unless you have five years of full-time experience in the field.