There is a theory that radical Catholics in the 13th century wanted a Crusade to Constantinople to reunite the Christians under the Church of Rome. It was never taken seriously by the Pope or the Kings, Emperor etc... They were more pragmatic in these matters. Although, any planned Crusade would result in a catastrophe for the Christian World in the Eastern Mediterranean. The Roman Invasion of Sicily in the early 14th century proved that the Romans were still a strong force not to mess with...
I mean, looking at the Albigensian Crusade it isn’t hard to picture the most zealous Catholics wanting to wipe out the heretics. It was a patently impossible idea, though—a handful of zealots couldn’t bring down Orthodoxy, let alone all of the varied flavors of Eastern Christianity.
The Roman invasion of Sicily did certainly put the fear of the Basileus into Italy and beyond—but in the long run I’d argue that was bad for Rome. That fear helped Giacomo Albizzi begin his unifying efforts in Tuscany, and in the long run paved the way for the unification of Italy as we know it. The Italian navy proved itself to be a huge thorn in Rome’s side on more than one occasion, not to mention its obscene economic power and influence.
Besides, it’s not like Rome held the parts of Sicily it conquered for more than two decades. They would’ve had better luck holding parts of the Italian boot proper, where IIRC there were some actual support for Rome among native Greek speakers.
If the Crusaders took Constantinople, the Turks would at least be able to take all of Anatolia, and then the Eastern Roman Empire is screwed whether it's Orthodox or ruled by a small elite of Catholic Frankish knights. Probably a Turkic Anatolian empire would combine Hellenistic and Turco-Persian attributes, raid the coast of Greece (they'd never actually conquer it though, as raiding it would be too profitable) and temporarily take Constantinople before the Italian age.
I dunno, the Romans still controlled Bithynia and beyond at this point. Total Turkish conquest isn’t inevitable even in the wake of a nasty collapse of authority.
If some kind of unitary Turkish Empire was founded, I’d agree with your analysis. The Romans could hold them out of Europe proper with their navy, not to mention support from other Orthodox powers like Serbia.