So, what if Alexius continued on, and relieved the besieged Crusaders at Antioch? What effects would this have? Would the crusade leaders still go back on their oaths, or would the land fall into Byzantine hands as promised?
The first effect would be probably a more important tendence of the crusaders to fulfill their oaths. Not by gratitude (even if it can plays) but because Alexios would have likely the control of Antioch and able to oversee places like Edessa, Tripoli, etc.
A second effect would have been to make the Anatolian roads more safe against the Turks : it would likely mean a sucessful Crusade of 1101, and therefore more Latins in Syria/Palestina (kind of a twist, I know). Places like Ascalon could even be taken in this decennial.
A third effect would be that Raimond of Saint-Gilles, that was one of the most important leaders of the 1st Crusade, would be more likely to be chosen as King of Jerusalem. See, IOTL he wasn't for several reasons (mostly because he was kind of powerful already) but being the only one not having gave his oath in Constantinople, it would gave him more room (and for his vassals consequently) in Palestine.
To resume : better byzantine control in Anatolia and northern Syria, more important Latin presence in Middle-East, Raimon as King of Jerusalem.
After 3 years of travel, hunger and battle I doubt these remaining elite men would think twice about slaughtering anyone they perceived (rightly or wrongly) as an enemy.
It is why Raimon of Saint Gilles, managed to enter in Jerusalem managing to let the Muslim garrison run free and actually trying to protect what he could of the population?
Look, I'm not saying they were chivalrous angels. No invading army is, critically in a quite violent era. But we could maybe cut a little the "they'll kill everyone in sight", situation being a bit more complex.