If you can cause the protein to be generated in the blood, why not also treat cells on the brain side of the blood-brain barrier as well to produce the protein and so protect the brain?
If you can do that, you can halt transmission to new people, and halt progression of the disease on both sides of the blood-brain barrier, it's about as close to a cure as could be achieved without actually destroying all the quiescent viruses.
Heck, it might actually cause less damage to the host than destroying all the quiescent viruses--the host cells are still sort of functional, but if you killed them all, their functions would fail completely.
See for example the antiviral treatment in research, which causes infected cells to die:
A worry with this treatment is that many nerve cells are infected with herpesvirus in some people, and simply taking these cells out may result in far more damage than desired. Better to leave infected cells intact and enforce dormancy, which protects hosts and prevents spread.