A free health system does not automatically protect against exploitation. The margin for exploitation remains, and potentially increases.
Under a publicly funded health system, the doctor or medical professional can still bill the government for unnecessary treatments. The government will then pay for unnecessary treatment from tax revenue. This revenue comes from the public purse, and the financial costs fall on the population at-large.
If the patient isn't paying for the treatment they are receiving out of pocket, they are likely to be less vigilant to unnecessary or duplicate treatments. Therefore, the margin for exploitation could increase.