Lab Housing Policy and Negative Gearing, page-37

Do they now. The majority of the properties become positively geared over their lifetime and even if they don't because the owner sells before they become positively geared overall the taxman does not loose due to the tax paid by cgt. And no the mortgage does not need to be paid off to become positively geared.
As for drawing down on the equity - that does not affect the initial property investment tax status as that equity goes towards the next property investment.