Wouldn’t it be better to remain loyal to the Orthodox Christian Byzantines than the Muslim Ottomans?
It's worth pointing out that there is often an imposition of a hard border onto Byzantine-Turkish relations that is simply untrue when we consider the sub-Constantinopolitan provincials. Even as early as 1077 the Byzantines used Seljuks as mercenaries, allies and placeholder garrisons for cities when their own resources ran scarce and this only increased as the capital's influence diminished.
By the post 1204 period we find Byzantines in Seljuk service in Konya where they established new lives for themselves, brought literacy and education to the bureaucracy and helped to expand the prestige of the Sultanate. There is interesting work in the thirteenth century as the Seljuks disintegrated and Nicaea had not yet reestablished a level of supremacy (not that it ever truly did after John Vatatzes) where we find communities of Byzantines and Seljuks living side by side. The career of Rumi is one such example and his poems give good evidence of tolerance and coexistence that belies the Holy Wars of the west.
When you come to the collapse of the empire to the Turks you are forced to confront the fact that Constantinople now meant little more than a religious centre to the post Byzantine Orthodox population and had long ceased as a political head. What Constantinople and the last Palaiologoi emperors tried to accomplish was a reunion of the churches of Catholicism and Orthodoxy that was unwanted by many who had made new lives and were happy to live under the Ottomans who protected them from banditry. The milet system of the Turks was in honesty quite lax upon the non Muslim populations while Constantinople only offered a betrayal of their faith. This was why the last Megas Duke Lukas Notaras issued his famous line 'better the Turkish turban than the Cardinals hat.'
Also worth mentioning the issue of Nicaean legitimacy as the continuation of Byzantium. The fact that in 1204 there were a number of different polities who claimed independence, often without the imperial title, was a major issue for the Nicaeans to combat and one which they could only force with military action. Epiros, Rhodes, Paphlagonia, Adrianople, Trebizond and what was left of Leon Sgouros Peloponnese state did not submit easily to the Nicaeans and it took several exercises in popular rule in the reigns of Vatatzes and Theodore II Laskaris to find a new exercise in popular control and legitimacy. This was eventually established in Asia Minor but remember that the Laskarids were deposed by Michael VIII Palaiologos and his son exiled and perhaps blinded. This isolated the ruling dynasty from the local support of Anatolias pronoians, the landed magnates who owed position to the old dynasty, and the attempted reunion of the churches aggressively pursued by Michael in an attempt to dissuade the West from another crusade upon Constantinople did so further.