The Crusades arose from the loss of Jerusalem as a counter attack. This is what motivated the West. Yes, the Byzantines were looking for help but the motivation for people to leave home and cross, for them, an enormous and scary distance was the loss of the Holy City. Urban II lit the fire by preaching a crusade but the people of the time really believed and wanted to redeem what was lost.
None of the Crusader leadership had very much respect for the Byzantines and they were definitely not there to help the Byzantine Empire. Indeed, in a later Crusade they conquered the Empire.
At the time the Middle East was full of warring princes who were, often as not, divided by ethnic and tribal allegiances. So, Arab against Turk against Kurd against Armenian and so forth (sound familiar?). The Baghdad Caliphate had, effectively, fallen so there was no more or less central Islamic direction while the Byzantine Empire was mortally wounded by the loss of central Anatolia due to a lost battle (Manzikert) and the Turkic “move in” after that loss (again, sound familiar?). Nearly 100 years later it took a Kurd (ironically) by the name of Saladin to retake the Holy Lands for Islam.
The Seljuk Turks, who were recent converts to Islam and fundamentalists, moved in and took all the area including Jerusalem from the Fatimids and became a threat to Constantinople, who would ask for military aid from Urban. The schism between Constantinople and Rome was in 1050.
The big battle between the Crusaders and Turks was for Antioch which lasted 7 months and finally captured by the Crusaders.
During the battle over Antioch, the Fatimids would recapture Jerusalem from the Turks, so after the Crusaders took Antioch from the Turks they took Jerusalem from the Fatimids.
I agree with your assessment that Runciman is the best source but the above article criticizes Runciman. I read it 20 years ago. Runciman uses western Christian, eastern Christian, and Muslim sources.