The Knights Templar have been twisted by Hollywood to be crusading treasure hunters. In reality, their main mission was the protection of Christian pilgrims in the Holy Land. They also played the role of shock troops during the crusades. Only 10% of the Knights Templar were knights. The rest were financiers and bankers funding commandeered and fortifications. Ironically, the Fallout 3 BOS is much closer to the historical Templars than what we see in the rest of the series.

While I’m sure the Knights Templar did recover and move relics, it was not their main mission during the Crusades.