The tiles on the shuttle's belly were the complete opposite. The main tiles on the belly of the shuttle were roughly 10% silica fibers, 90% air. Think very low density styrofoam, except that it can be heated to glowing temperatures without losing its properties. This was actually the really cool demo that I saw. The person giving the demo heated it with a torch until it was glowing yellow/white, then picked it up with his bare finger tips. Because the thermal conductivity of it was so low, it could be handled (with care) with bare hands.
For the OP, the point of the thermal protection system was precisely the opposite of being a heat sink. It's entire purpose was to insulate the shuttle against the heat that the belly was exposed to during re-entry. Contrary to popular belief, the majority of heating during re-entry was due to compressive heat (think diesel engines, boyles law and all that), Not friction. Basically the shuttle would compress the air in front of it, causing it to heat up to plasma type temperatures, which was then transferred to the body of the shuttle through convective heating. As such, the best way to deal with it was just to insulate yourself, and wait for the high temperatures to pass.