If there were any market incentive whatsoever for that, it would already exist. The FDA does not have magical anticompetitive powers that make people care less. We have private and public postal services and railroads operating alongside each other.

The best case scenario is that there is enough market will to allow a lab to monopolize the market, leaving you with the same problems as today but with even less standing in the way of fraud, like companies just paying for good reviews. The realistic case is that no such testing would happen, just like it was before the FDA.