Several points here:

(1) Already in the early Middle Ages, pilgrims to Jerusalem report seeing an important relic (tangible object/body part associated with Christ/from a saint, like Marie d'Oignie's fingers or Christ's foreskin from his circumcision) in Jerusalem: the "Holy Chalice," that is, the cup Christ used at the Last Supper. The idea of the Grail, which will eventually be the cup Christ used at the Last Supper repurposed by the apostles to catch his blood underneath the cross, on the other hand, derives from the later 12th century Arthurian tradition. Chretien de Troyes mentions a mysterious "graal" in Perceval, which is more of a serving platter; Joseph de Boron and later developers of the Arthurian romance tradition will define this into our Holy Grail. All of that literary development, please note, significantly postdates the First Crusade.

(2) Discovery of relics strewn about the Holy Land did become a major part of Crusader culture. Scholars talk about a "steady stream" of relics flowing back to western Europe--shards of the True Cross, heads of John the Baptist. The most famous case is actually one where people ultimately concluded the relic was false. A French soldier named Peter Bartholomew claimed to have discovered a lance in an Antiochene church on the advice of a spectral St. Andrew, which he assured the Crusader army was the lance that had pierced Christ's side on the cross. Leaders were skeptical. They set up a trial by ordeal for Peter Bartholomew to undergo--if the lance were the Holy Lance, God would prove it through Peter's survival. He did survive the initial ordeal of fire, but subsequently died of his injuries. The lance was declared a fake. However, all this relic brouhaha was an ex post facto success story, not an original motivation.

(3) I'm not sure why you say a legend created "to give the Crusades a purpose." In the rhetoric of the time, the Crusade(s) had an explicit purpose: to recapture the Holy Land. Scholars will debate the trappings--the Crusades as an act of Christian love, the extent to which the West was actually motivated by helping Byzantium (defensive war?) versus itself, how much was spurred by expansionary desires/socio-economic motives, Crusades as pilgrimage, Crusades to protect pilgrimage, recapturing the Holy Land as foundational to proper order in the world--but "Latins conquer Jerusalem" was a clear and sufficient mission statement to people at the time.