1. Otto III was himself the son of a Byzantine princess, and that did little to bring the empires together apart from sustaining a peace under which the Germans stopped attacking Byzantine Italy. Why would a son of Otto III and Zoe be any different?
2. The Byzantine emperors already did see the HRE as "legitimate." They reserved the formation "Emperor of the Romans" for themselves, but since the 9th century the emperors in Constantinople had little trouble recognizing the Frankish/German kings as emperors too (just not emperors of the Romans). You don't send an imperial daughter to marry into the ruling family of a state you see as illegitimate.
3. Byzantine succession was not strongly hereditary; to win the throne you had to actually impose yourself upon it, and in the event of a succession crisis in Constantinople it's almost certain that a Byzantine general or other strongman is going to seat himself in the city before word of the crisis can even reach the west. If a son of Otto III wants to win the throne he's going to have to do it the old fashioned way, by actually invading and conquering the empire, and considering that Otto III couldn't even manage to keep control of Rome I don't see that as a very likely military feat. "Strongest claim" is meaningless compared to "strongest army" and "strongest support by the native elites."
4. The Germans and Byzantines really have no ability to aid one another in each others' home territories. The best possible outcome of a closer alignment between the HRE and the Byzantines at this time would be if they can cooperatively prevent the Norman conquest of southern Italy, in which case the Byzantines are spared from a lot of 11th century headaches and are going to be in a significantly improved position by the time the Turks roll around (though I can't say whether it will make any difference).
5. The Germans never much cared for the habit of the Ottonians to decide they were really "Romans" and rule from Rome. Otto III didn't live long enough for this to be a problem, but if this becomes a long-term thing, Otto III is probably going to face some serious pushback from the Germans back home, and his theoretical son is going to have the same problem. If the German emperor decides to invade and force himself upon Constantinople, I have my doubts about whether the German nobles are going to support him. If he goes anyway they might just decide to put a "real German" on the throne instead (like the historical Emperor Henry II).
6. Let's assume this scenario actually works and a Hellenized Ottonian is on the throne of Constantinople. What of it? Brothers fight each other all the time, historically speaking, and cousins even more so - there's no reason to assume that there would be a permanent or even long-term alignment between the two empires any more than there was IOTL. Aside from some sparring over southern Italy (which becomes irrelevant after the Norman conquest), they are too far removed from one another to have many common interests or foes (again, apart from the Normans).
7. The division of Christendom into Latin and Greek spheres was already an accomplished fact at this time, even if it wasn't made "official" until 1054 or so (and even that was viewed as a relatively minor event at the time). I doubt the mere fact that they guy on the throne in Constantinople has Ottonian blood would have much appreciable effect on this situation.