It’s pure speculation but if instead of sacking Constantinople and killing all of those people West made an alliance with East, Byzantine Empire could have survived, albeit losing substantial territories in the East. The situation of Eastern Romans was bad with them losing a lot of their eastern provinces but they still had some fight left in them.
The Byzantine Empire was faced with a conundrum. It needed political allies and military support to be able to see off the threats of the Seljuks, then the Mongols, and finally the Ottomans. The major Latin Christendom powers were hesitant to give that kind of consistent alliance to the heretical Orthodox/Catholic split, and many Byzantine emperors did seriously consider the idea of giving up the Orthodox to obtain that support. However, the emperors who did so faced the severe wrath of their core population, driving rebellions instead.
The Byzantines were effectively forced into a mode where they had to choose between internal and external security, which is a large factor of their terminal decline. By the Fourth Crusade, the Empire was pretty much in a state where the surrounding powers had more reason to wait to pick at its corpse than to help it survive.
I've thought about that a lot too. It's possible, but I think that thread failed when the re-conquered cities were turned into the Crusader States rather than being returned to Greek control. If anything, I feel like the Siege if Vienna would have happened at the walls of Constantinople instead and history would be more or less unchanged.
You mentioned Siege of Vienna. Remember that it eventually failed and it culminated with Battle of Vienna when Ottomans were completely defeated and soon after driven out of Europe. I would have thought that had the battle happened outside Constantinople instead of Vienna the history would have been changed completely. Vast parts of Ottoman Empire would probably be part of EU now.
