Yes, any public or private school can accept vouchers. However, that voucher does not fix the interlocking issues that occur. Take student A. She lives in a poor achieving school district in rural America. Her family decides that they want a voucher to go to a better school. They get their voucher and find a school that will accept them. That voucher has a variety of effects. First, it impoverishes the school the child leaves. Second, it does NOT pay for the transportation of that child to the new school. Third, it removes from that child all local support.
For a clearer example, take my school district. Rural Colorado. If I didn't want my daughter who is 15 to go to the local high school, I could ask for a voucher. The next nearest school district and nearest high school is 17 miles away. Due to the fact that the high school is also rural, and in declining enrolment, they have a waiting list for non-district students.
Next high school and district. 23 miles away. It is in a category known as Turn Around which means it's scores are failing. Therefore it is out of the running.
Next high school and district. 50 miles one way. Three high schools, Some are in Turn Around status.
To be honest, our solution was to homeschool her and use KhanAcademy.org to fill in where we could not. Even with a voucher, we would have been unable to afford the transportation to an appropriate high school. Vouchers are simply not practical in Rural America where school districts are few and far between and the poverty level is a deciding factor.
As to why I'd put money into a poor system... because it is all we have. We have to make it work to the best of our ability and robbing it via a voucher system will not improve it.
Oh, and the link would not work.