I was under the impression that a special vascular netting has been created to temporarily stop blood and other clots from making it to the brain through such procedures. Later, the netting can be removed from the arteries after the body has recovered a bit from the operation.
Surely this hasn't been commercialized yet, but it does point toward a possible solution.
As for blood stoppage to the brain, I would imaging they keep that under pretty tight control and that it is supplied enough to not be a concern to the patient.
Around the time of Clinton's surgery, an alternative medicine newsletter to which I subscribe discussed the advisability of this surgery. They said that the public description of Clinton's illness suggested that a non-surgical approach -- diet, exercise, medications -- should be tried first, especially because surgery carries the risk described in this article. Apparently, even with the best of care, surgery may leave you with a mental deficit for periods ranging from days to forever.