Getting a work visa may be burdensome, but not impossible...
If your future US employer is willing to be your sponsor or if you are very lucky to obtain a green card. This will only happen if your employer can't get skilled employees in the States. Best chances I guess, get married with somebody living in the States.
Atlas and other US freight carriers take many Europeans
I speak to pilots from Kalitta, Atlas, Polar and Southern from time to time, who are operating in and out of Europe. Have so far only spoken to US-citizens. (non of them Europeans). If what you are saying is true, US carriers can (especially FEDEX and UPS) expect to get inundated with applications from European pilots.
As I said, I have not heard of European pilots being able to work for US carriers with European bases. I encourage the US carriers to do so, that can only improve the terms and conditions of the European freight carriers.