People in poor countries working for people in rich countries is how wealth is equalized. That's how developing countries develop and how developed countries stagnate. So the existence of an international market for this is probably already part of the solution. After they have enough wealth, they can give themselves free healthcare and education without being a basket case forever depending on foreign charity.
Your argument is orthogonal to free healthcare and education being offered by the state. You don't need to be an immigrant to be poor in your country of birth, such as the US.
