It doesn't matter what you do. Your healthcare is either going to be the subject of profit based motives (private industry), or budget based motives( read: election votes). We would have changes to the UHC system every time an election came up.
Our system, flawed as it is, is a mixed system of pooled risk. Some sectors (poor and elderly) are covered or have the potential to be covered (if they choose) by government programs. The young and typically healthier partake in a private insurance industry pool risk (for the most part). So we already have both here.
What people in this country want is top notch medical care and services extended to everybody. The problem is, nobody wants to pay for it themselves.