Was not the title of Universal Bishop much sought after, the Bishop of Rome winning it because he had the largest number of adherents?

No. Whatever abuses arose in later times, the early saintly Popes, nearly all of them martyrs for Christ, were not the men to seek after office, and dignities which they knew to be spurious.

Radio Replies Volume 1 by Rev. Dr. Leslie Rumble MSC and Rev. Charles Mortimer Carty