I think the main problem is the immigration system. Everything runs very slow in dealings with the INS. Trust me, I'm currently going through it. An H1b or Permanant Residency application handled through a lawyer will cost a company about $2-3000 and take anywhere from 8 weeks to 4 years (depending upon your eligibility status).
Even if you can find a company prepared to do this (and there are ones out there), some further hurdles exist. For a "temporary work permit", a Labor Certification is required which basically states, is an American able and avilable to do this job? If so, you won't get it. Exceptions to this are medical staff or IT people.
As for a shortage in the pilot hiring market, there isn't really one, except at the instructor and maybe some cargo pilot levels. Regionals simply lower their hiring mins and hey presto, a new load of eligible new-hires. As mentioned on here before, most majors have an average of 7000 current resumes on file!
It has been said before, if you're single and looking to marry...there are plenty of american men and women here! (not me, I'm already taken :-)