>Retaliatory
i mean you admit to the Islamic consolidation in Anatolia, the eastern Romans were smashed at the battle of Manzikert
and also the destruction of the church of the holy shroud by the fatmids (yes it was targeting the seljuks) some generation or 2 before.
the only way the pope could get that relief to the easterners was through this religious wording targeting the aristocracy because asking or advising kings is like pulling teeth, see is fucking successor and why his successor couldn't call a crusade.

>but it wasn't lead by the emperor and they seized ground
yes, those things aren't necessarily incongruous, the pope lacked the ability to control these groups generally, even on their own turf, see the fucking Normans.
the pope decided to levy this army without any property way to control it and like the drunken idiots they are they surged disorganized through europe, randomly pillaging until they got to the holy land.

also do you really think any latin is going to defer to the emperor? particularly given he was effectively de clawed and was begging for help after getting his arse kicked

"Retaliatory"

Yeah this is why they took place like 400 years after the initial conquests.

And this is why the Crusades were led by Western European Catholic Princes rather than Greek Orthodox Byzantines, and made them into their own kingdoms rather than give the territory back.

Seriously that would be about as "retaliatory" as let's say Germany invading the US to reconquer New York/New Amsterdam … oh not to give it back to the Netherlands but to keep to for themselves.