This based on the idea that free markets are better at operating efficiently.
Thing is that the ultimate goal of any enterprise is wealth extraction.
The most direct such extraction is via racketeering.
But baring that the next best option is to offer a tangible product or service others are willing to pay for.
But note that the goal is not to offer good product or service, it is wealth extraction. Thus one will aim to offer the most minimal product or service one can get away with at the highest price.
So no, i don't think there is any reason why they should fear that the US system is being gamed to introduce public health services. It is being gamed to maximize wealth extraction by everyone involved.
On the other hand, ruining existing public health services to make private equivalent look good would directly lead to increased wealth extraction by those involved with the private health services.
It's actually not an uncommon opinion in the UK, at least among the people I know. In fact, Thatcher herself introduced much of the bureaucracy into the NHS in the form of the "internal market".
In the US they are literally doing this right now with the post office. In the UK they did it with the rail network.
It's a standard part of the privatization playbook.
I dunno, maybe healthcare is just hard to get right, whether run publicly or privately. Maybe we still know very little about how to treat and fix the human body, but rapid progress in other areas of technology and the numerous developments we have made in healthcare lead us to think otherwise.