You're claiming that "totally deregulated private doctors" are better in terms of cost and service than any other system, but you can't give a single instance of a first world system that does this. More than that, you seem to be cherry picking the best of the examples you did give, which is not really helpful since the best in the U.S. (if you have a good insurance plan) is as good or better than these other countries.
India has some good healthcare in the major cities and major problems in the rural areas where many of its people live. Mexico is a public-private hybrid, like Australia and many other countries. Again, you can probably find excellent care in some urban areas which likely cater to medical tourists, but they do not match up to more advanced countries as far as high tech care goes.
If you can't give an apples-to-apples comparison for what you're advocating, then I don't see how helpful your example of "private being better" really is.
India has some good healthcare in the major cities and major problems in the rural areas where many of its people live. Mexico is a public-private hybrid, like Australia and many other countries. Again, you can probably find excellent care in some urban areas which likely cater to medical tourists, but they do not match up to more advanced countries as far as high tech care goes.
If you can't give an apples-to-apples comparison for what you're advocating, then I don't see how helpful your example of "private being better" really is.