Hi Satya,
Yes, they would be suggesting IVF because of your age. And because after doing all the preliminary blood tests, ultrasounds and lap and d&c, the actual process of IVF can help clarify whether you have any other issues going on and also it sets up the best environment for pregnancy ie. they can put the "best" embryos back in and you get progesterone support to make sure the environment is right in the TWW.
We TTC'd naturally for 3 years between the ages of 41 and 44 and we already had one child so in theory my only problem was age. I did IVF at 44 and got pg. on the first cycle. (but miscarried, but at 44 my chances of miscarriage are over 40%).
So I would go for the IVF, if I were you and I would be ringing around to make an appointment as the first appointments are often about a 10 week wait.
On the public / private thing. The IVF treatment cycles are not covered by private health insurance. Only your egg pick up, as it is done in hospital is covered by PHI. I think the royal womens is where you go to get the egg pick up done for free. I am not sure how much you have to pay for the treatment cycle.
Monash prices are reasonable and you would probably end up about $2500 out of pocket including the egg pick up and treatment cycle if you went through them. the money thing is confusing so ask me for clarification if you want. hopefully someone who has gone public through royal womans may be able to help
good luck