You said the magic words: "in Italy", where the Papacy was a) a temporal power and b) it could involve itself directly in the diplomatic relations of the various Italian states, who all acknowledged its authority and themselves tried to win its backing. But the topic of this thread is not Italy, or Western Europe, but the Balkans and the Levant; and here the writ of the Pope did not carry far. Even the papal interdictions against trade with the Muslims were frequently disobeyed, land the inability of the Papacy to organize any effective or long-term Crusading movements post-1204 is a recurring theme. What 'crusades' and holy leagues there were were more the result of local Latin powers (chiefly Venice) coalescing with Papal sanction, rather than initiatives deriving from the Pope himself.
All political entities have power that surges forth from a heartland of power. The Roman Republic projected political force from a location that they firmly held, so too was the case for the Papacy. When the Papacy was lords over Italy and not small entities, as they were in 1415, it permits them to more thoroughly back their temporal authority over England, France, the Empire and so forth.
For instance, Peter Anascarid king of Castile was deposed by Urban V. Urban V, had from his predecessors asserted a relatively powerful realm in the Occitan region, becoming its strongest feudal lord, in a way mimicking the old County of Toulouse in power. Nevertheless, Urban V with a powerful realm was able to issue interdiction upon Peter of Anascarid and then took to declaring Henry of Trastamara king of Castile and sending him with an army of Crusaders, not different from the ones that Urban V had sent in the Balkans as part of the Savoyard crusade or of crusaders sent to Egypt that sacked Alexandria.
When the Papacy held their allodial lands, it possessed the power in essence to carry weight directly outside of legal and its otherwise indirect holdings (namely, the 35% of Europe under his demense).
Interdictions of trade were applied by the Papacy with reservation for the realities of the matter. In otherwords, the Papacy reserves rights to enforce all legal renderings, it is not a ruling that the Papacy held as being constant or obligatory. (this is a major distinction; for in Sharia and other laws, laws are obligatory and not determined by context or levels by which one enforces them) When Innocent III issued said rulings, he made such with the context that simply, should the Papacy declare a trade relation invalid, this is permitted by custom. However, this does not mean that Innocent III was ruling that all trade with non-Papal mandated states was forbidden, quite the contrary. And even so, Venice was technically not entirely a Papal subject in the same way that say, France was in the opinion of Innocent III, so rulings and interdiction apply differently to them.
I disagree, Papal crusades after 1204 were very effective. None would argue the central factor that the Papacy held in crusades to depose Latin monarchs, to push towards greater colonization/settlement/expansion in Iberia and the fundamental role that the Papacy held in the blocking of Mongol adventurism into Europe. Later, the Papacy was launching crusades against Latin and Muslim states consistently with varied success. Simply due to not always succeeding, does not deny the powers that the Papacy held, it was not a power to be trifled with as Venice would learn, surely.
Surely, if we permit the Papacy to resurge in Italy, Venice will be an after thought as they were in otl, an inferior power in comparison to the 'big fish' of Europe;the Papacy, France, Empire, England, etc... It is difficult to even conceive that the Italian states ruling parts of the Balkans would be able to maintain Balkan integrity when the Papacy is handing their land titles to Hungary, the Empire and waging overt war with them in the peninsula.
Regarding Byzantium, they have an incentive for friendship with the Papacy. There is little reason for them to not at least make friendly terms with them and enter mutual assistance.