So it isn't illegal?
The reason why it's an increasing practice in Europe is because it frees the employer from legal responsibility. Labour rights are coupled to a formal employment contract between the air carrier and the pilot. Now pilots create and register a firm in their own name that signs a professional contract with a third party crewing company. Being in effect a contract between two contracting agencies it frees the air carrier from all employment obligations. It also robs pilots of the right to strike, shifting the balance of power heavily towards the employer and effectively empowering them to dictate employment standards altogether.
The competition in the US is harsh, just like in Europe. I would guess this would be an attractive option for american air carriers if they had the opportunity.