We do pay American tax dollars directly overseas it's called foreign aid.
Of course. Individuals on the receiving end do not play a role in deciding when/where that money is spent though. Foreign aid is something entirely different, it is something decided upon by congress. An individual illegally crossing the border with the express purpose of seeking treatment for a pre-existing medical condition is an entirely different thing.
When you mention that 'passport control came out of World War I'. This is incredibly disingenuous. The politics of pre-WWI Europe can not be compared to the modern day. You are talking about an era where ideas that would be considered radically nationalist were completely normal. You may have been able to cross borders, but your ability to integrate economically into a new country may have been limited. Not to mention the fact that the entire economic landscape of this point in history is not comparable to our current era. There was no globalism, global trade and global industry would have been extremely minimal compared to today.