I don't quite get what you mean about running water washing away blood. From what I understand running water had to be used by the priests for purification before they served in the Temple. I could be wrong about that, but, at any rate once running water is put in some sort of container and transported some place it ceases to be running water
and water doesn't run up hill if that is where the Temple was really located on what people are calling the Temple Mount.
The only source of running water was the Gihon Spring and that is below the Temple Mount.
In his book "The Temples Jerusalem Forgot", Mr. Martin makes an interesting observation:
Pg. 467 Once this southern location for the Temple is recognized, an historical account by Josephus now makes sense. He said that before the war, Agrippa the Second would customarily recline at dining in a veranda room at his palace with his friends. From that spot they could look at what was happening within the Temple courts. For Agrippa to see that interior of the Temple the elevated part of his palace (where he would dine with his friends) must have been in the Upper City and at an elevation higher than the western wall of the Temple. When the religious authorities heard that Agrippa was entertaining his friends in such a manner (viewing the religious activities within the Temple courts), they decided to prevent this by constructing a new wall on top of the western colonade. 699(Antiquities XX.8,11). This additional height obscured Agrippa's view.
Look at this incident carefully. In no way could Agrippa have seen inside the Temple courts if the Temple were located over the Dome of the Rock within the Haram. That area in the north is much higher in elevation and would have been far too elevated for Agrippa to look over the Temple walls into the courts where the worshippers assembled. This again shows that the Haram (Tample Mount) can in no way be considered the site of the Temple of Herod.
700 I am indebted to the Israeli architect Tuvia Sagiv for ponting it out to me on his Web Site. Indeed, with the actual Temple being located 600 feet south of the southern wall of the Haram, the observation of Tuvia Sagiv makes even better sense
(War VI.2,6).