You CANNOT become a doctor without a medical degree. Getting a PhD instead of an MD might be a good option if you wanted to do medical research without ever caring for patients, but you cannot treat patients without a medical license (which you will not get without a medical degree).
Getting into a PhD program is no easier than getting into medical school. I am not sure though why you think getting into medical school is particularly difficult for international students. The only obstacle is financial. If you have $300,000 sitting in an escrow account, most medical students would consider you for admission like any other student.
Are most graduate schools centered upon the sciences really fully-funded?
Most of the decent ones are fully funded, but you'll also find plenty of programs that are not.