I think the summation of this is correct
The When do you start it :
a. 1025 + -- Strong army -- massive economy -- (Basil did not collect the hearth tax for 2 years) - Invasion of Sicily keeps Normans out of the Med. / If a good emperor or if George Manices takes over then you prob can conquer down to Jerusalem (1/2 or more of the population is still Christian) and the coast. If you take Damascus and other other inland cities you have excellent defensible borders. So the Byz military do not face crusades - weak Muslim states strong economy which will only be enhanced by the conquests . So you need to face the Mongols and then you have really no major foe in the future (Tammerlane?)
Egypt was also not-quite-overwhelmingly majority still Copt by around then, still. Exact figures might or might not exist (seriously, I don't know), but I've heard 70% or more bandied about before.
Of course, the Coptic Christians aren't necessarily going to be on the greatest of terms with the Orthodox Byzantines. They're going to be used to the extensive self-government that relatively tolerant Muslim government had given them, an Emperor who rolled in and tried to change that is going to face some problems. You'd need an Emperor who is going to understand this to take Egypt (and the Levant, for that matter), and be willing to make compromises against what are probably going to be some very passionate wishes coming out of his own Church's hierarchy.
And not only that, but you're going to need Emperors to KEEP understanding this for quite some time. An Emperor who lets zeal get the better of him who tries to push on the 'heretics' in Egypt and Syria is going to have some of the same problems a recent conqueror would have.
I actually like the idea of the Romans being able to take the old Diocese of the East sometime in the Middle Ages (Nikephorus sounds the most plausible: He had the means and intention to do so, just got killed before he actually could) but losing it during a period of instability at home. If the Turks roll off the step to face a powerful empire that has only loosely held the East for a few generations, they might still have the ability to remove that territory for themselves. Maybe have portions of the area taken and lost again until something analogous to the Mongols comes around and causes a real crisis, breaking the cycle and leaving the Byzantines locked out of the East until it's ready to go all Gunpowder Empire everyone around them.
Syria, Palestine, and Egypt retain much larger Christian populations because Muslim dominance is much less continuous, so by the time develop somewhere begins to move on into modern economics, the Byzantines are an Eastern Mediterranean powerhouse who just has to play some catch-up, rather than being permanently relegated to being Greece + Bulgaria + Turkey.
I don't really buy into the "surviving Byzantium nerfs he Renaissance" concept. If that were true, we'd have to think that an Ottoman conquest of Italy would spur it on even faster, as Italian thinkers flee to the rest of Europe.
Byzantium's going to maintain strong trading ties with the rest of Christendom (stronger than the Turks, for sure), so whatever philosophical, artistic, and technological developments you associate witn the period will likely occur on schedule. Maybe even earlier, with Greece and Anatolia thrown into the mix.
Well, the interesting thing is that a lot of the intellectual creativity that characterized the late Byzantine period in Mistra is probably a result of the declining central power. A powerful Empire might well just stick with the old schtick of 'keep copying Antiquity, do nothing new, nothing new can be better'. It often takes crisis for people to look for new answers and deep crisis for them to not be slapped down for questioning the old ones.