Yip that is correct, to a point. Now that there are many airline jobs available in US why go outside and take the risk of entering something unknown.
Still a lot of the VIP pilts I have met flying big jets in Middle-East are from US.
For me... Well if I had a greencard I would not have anything against working in US as long as I could come home to EU as often as possible. So I would rather work here or somwhere else.
JJ