Trying to back into it. Word parsing is such a terrible argument. That doesn't say that right applies to all Americans, it just says that you can't prevent someone from voting by charging a poll tax. Why even say the part of the poll tax if every American has that right? Just say, every American has the right to vote. Period. That covers everything else.
If I say your right to drive cannot be restricted by the State you live in, and I mean you have a right to drive, I don't need to say the part about the State you live in. That you have to try to back door through word parsing is pure desperation. All you pointed to was the word "right," which they did not define anywhere as meaning every American can do it. Every other right is specified, they don't just say something is a "right." They say WHO has WHAT right. All you did is word parse the word "right" itself.
What they did not say is who has that "right" or what "right" they have. They just said voting cannot be rejected for reasons X, Y, or Z.
Again, if by "right" they meant every American has that right, why did they not just say that? Why bother with saying it can't be denied for reasons X, Y, Z and never actually bother to say that right applies to everyone?