I don’t think it would be possible to work in the US without first getting training there, although you wouldn’t have to start at the beginning. The typical route to becoming a doctor in the US is a four year pre-med degree, four years of medical school and then a residency program. The length of the residency depends on your “speciality” - so Family Practice is two years, Internal Medicine is three years, OBGYN is four years, General Surgery is five years and so on. I’m not sure how long An(a)esthesiology is (four years?) If you wanted to specialise further after residency, you’d go on and do a fellowship.
As a foreign doctor, assuming your basic medical qualification is recognised, you’d still have to go through residency again. During residency you are working as a doctor (under supervision) but you are not making the big bucks. More than enough to live on but not the fabled high US doctor salaries. Once you complete residency you find a proper high paying job!
Residency programs are competitive and as a foreign doctor you are more likely to end up in a less competitive and therefore less desirable residency. You could try applying to a US med school after your first degree but I think the chances of getting in are very, very slim as med school is also very competitive and as you pointed out, they take on very few foreigners.
If you are entirely self-funding and starting from the beginning, ie doing your first pre-med degree in the US, I think you are looking at a total cost of around $500,000 or maybe a bit more. This is one reason US doctors get paid a lot - they have massive student loans to pay off!