I doubt single-payer will cut costs in half.  It might be a substantial savings, but as long as the marketplace determines the cost of healthcare, and people equate paying more with as the ad says "a chance to live longer" the market will be driven to provide the most, not the least, expensive treatment possible.
From what I have seen, only countries that have managed all aspects of healthcare have costs about half that of our system. These countries don't deny healthcare to people who want to buy their own. So the so-called fear of lack of choice is not based on fact. What they do is protect the taxpayer money used to pay for public healthcare by weighing cost with benefits. For example, if they can't pay for new prosthetics for children, they might not fund new hips for those over 85. 
We can't pay for every exotic treatment. And choices about what we fund are already being made. 
Still, single payer is better than what we have.