In the US direct entry captain is very rare. Also, most pilots have resigned themselves to a career at a regional airline. Those who make it to a major are few and far between. So you should see yourself starting out at a regional if you can make it to the US.
Also, you will probably never make it to a major airline in the US if you don't know anyone working there. That is just fact. If you don't know anybody at Southwest who will write you a recommendation, then your chances are next to 0.
If you have a green card, you are one step ahead. Be wary of those who say that you can work at Southwest since you have a 737 type. Yes,
Southwest requires a 737 type, but type ratings from foreign licenses don't transfer to FAA licenses usually except for private pilot priveleges only (unless you have a Canadian license I think but I am not the expert in this field).
Yes, foreign pilots get hired in the US, those who already have right to live and work get treated equally, but when thousands of pilots are on furlough like now, American pilots and the like are going overseas to find work.
If you just got a green card in the lottery recently, congratulations. you just need FAA Certs and you can join the queue. Best of luck