This is where you are mistaken. First, to be clear, these were Eastern Catholic churches as well as being orthodox (in other words there was no difference at the time, and in fact the Orthodox have changed very little since).
The Holy See (by which we know you must mean the bishop of Rome, but they are in fact all holy Sees) was not ‘running’ things at the time. This was still a developing idea in the west about the role and function of the bishop of the city of Rome.
The Gregorian Reformation (which was to launch many changes in the role and function of the Papacy) had barely begun when Cardinals Humbert and Frederic went to the East. The Gregorian reforms were to continue in the west for several decades. Up until this point, the ‘Pope’ was not running anything in the east. The bishops of Rome had a limited involvement in the eastern churches which usually manifested when the man was invited or asked for his opinion (which was respected). He had no administrative function in the east.
In other words for the eastern churches he did not hire/fire bishops, erect dioceses, establish missions, regulate liturgies, write canons, canonize saints, publish/approve catechisms, call councils, approve the foundation of religious orders, etc. He was not “running things”.
In many parts of the western church the Pope’s authority was also limited even beyond the tenth century, but that was to change later.
So when someone says the “ortodox church didn’t like how the holy sea was running things”, there is a likelihood that these people (you are not alone in this idea) are projecting back in time a modern form of Papacy that did not exist then.
What might reasonably be said (and I would not basically argue with) is that some new ideas about the Papacy, the role of the Pope, were emerging in Italy at the time and the eastern orthodox Catholic churches were not in agreement with them.
If the Cardinals Humbert and Frederic were rebuffed for trying to impose this new authority in the east, they were rebuffed precisely because these were new ideas to the east. In other words, the Popes had never before “run” things in the east, although they appear to do so today in the Latin and eastern Catholic churches.