Actually the original plan was to put the Crusaders under Alexios, and the leaders of the expedition did swear fealty to him - and the two collabored in helping Byzantines take over many locations in Anatolia such as Nicaea. Ultimately the plan was that the Crusader army would take over the Holy Land and then return it to Alexios - but this didn't happen and the Byzantines were pretty pissed at their betrayal.
So it's not that the Orthodox didn't want to help the crusade, it's that the Crusaders didn't really want to help the Orthodox.
It would be interesting if it were possible for a human player to mend the schism the other way around: crusade for the holy land and then swear fealty to the Byzantine Emperor in exchange for his conversion to Catholicism, making Orthodoxy a Catholic heresy.
Depending on circumstances, it could even be the smart decision. Taking Jerusalem is one thing; keeping it is another. With a strong ERE's protection, Jihading for it would be a much taller order.