I think it’s a legitimate concern, and I would bet that most people who advocate school choice would like it to be available to every kid regardless of income. But keep in mind that school choice does not present a net increase in money spent on education for kids on the “welfare reservations” as you put it. The districts in those areas are already supported by fed/state/county aid; vouchers simply take that same pot of money and allow it to follow the kids to wherever they go to school.
I understand that, but when I am the municipal/county/federal taxpayer it means I am providing vouchers so gibsmedats can go to better schools than my own children (even though my district is better).