The best argument for private healthcare is that it creates competition which drives prices down.
If you are being charged random amounts of money for the same stuff based on seemingly arbitrary factors then that would suggest this is not happening as it should.
The current healthcare model doesn't promote competition. Providers must inflate prices to cover costs, since they know they will be reimbursed by insurance based on the insurers pricing model, not their own.
Private healthcare would have to start providing actual costs to insurers, and insurers would have to accept that those costs include losses from people without insurance for competition to take effect.
