Others have of course addressed this as well, but:
1) The Orthodox Church, more accurately the Orthodox Catholic Church, considers itself to be the Catholic Church, so no; no Orthodox Christian would concede that point. From a narrow theological perspective, the Orthodox view is that it's the one true Catholic Church; the organisation based in Rome is a schismatic hierarchy that broke away from, and is no longer in communion with, the Catholic Church. From a fairer-minded historical perspective, what we call for the sake of convenience the Orthodox and Catholic churches gradually evolved in different directions following the collapse of the Western Roman Empire, and then broke away from each other in the 11th century. Both can realistically claim direct apostolic succession from the imperial church of the Roman Empire; along with the Oriental Orthodox communion and the Church of the East, there are four denominations that can make that claim with some degree of confidence.
2) There were Popes of Rome before the Great Schism, just as there were Popes of Alexandria, Patriarchs of Antioch, Patriarchs of Jerusalem, and Patriarchs of Constantinople for centuries before the Great Schism. The Pentarchy is enshrined in the mutually recognised Ecumenical Councils, and for centuries the Bishop of Rome was subordinate to the Roman Emperor in Constantinople (a period covering every Pope between Vilgilius and Zachary, and still known as the 'Byzantine Papacy'), so I'm not sure what point you're trying to make here.
Historically - and history and Tradition aren't always in concord here, alas - there's every reason to believe that the idea of a single recognised bishop ruling over a specific diocese originated in the East before it was established in the West (see, for example, the epistles of Ignatius of Antioch), while the concept of Apostolic Succession evolved first in the West (see the epistle we now know as 1 Clement); they were then combined to form the commonly accepted definition of a monarchical bishop whose legitimacy rests on Apostolic Succession.
It would be fair to note that the status of the Archbishops of Constantinople and Jerusalem as full members of a defined set of senior patriarchs developed after those of Antioch, Alexandria, and Rome, but then we have much better historical evidence of monarchical episcopal structures existing in Antioch and Alexandria before they existed in Rome; the term 'Pope' was used to describe the Bishop of Alexandria decades before it was applied to the Bishop of Rome (and is still a title of the Coptic and Orthodox patriarchs of Alexandria). The early Roman church - as with many Christian communities - was most likely initially led by a community of presbyters rather than by a single recognised bishop, though with some figures (for example Clement) enjoying a certain level of pre-eminence within that community. The idea that there's a single unbroken line of bishops from Peter through to Francis is a lovely tradition, but historically problematic.
The bishopric of Byzantium / Constantinople is held by tradition to have been founded by St Andrew the Apostle in AD38. However, like the bishopric of Rome, it likely took some time to emerge as a monarchical bishopric, and it's notable that none of the epistles of Ignatius of Antioch are addressed to what was then still Byzantium. Like the bishopric of Rome, it emerges clearly as a verifiable historical institution in the 2nd century. It then takes on a new significance once Constantine I re-establishes Byzantium as New Rome - more commonly Constantinople - becoming the Archbishopric of Constantinople under Alexander (314-347), and then being recognised as second in status to Rome in AD 381 via the third canon of the Second Ecumenical Council; this was confirmed by the Council of Chalcedon in AD 451 (much to the chagrin of the bishops of Antioch and Alexandria), at which point the concept of a Pentarchy of 'patriarchs' takes firmer root, and the title 'patriarch' becomes more formally used. The Orthodox Church, however, counts Alexander's predecessor Metrophanes (306-314) as the first full Patriarch of Constantinople on the basis of an unverifiable tradition that Constantine granted him the right to use the title. Many early bishoprics considered themselves 'patriarchates', and not all survived with that status recognised and/or intact - see the interlinked histories of the patriarchates of Grado, Aquileia, and Venice for examples in the Western Church (Lisbon doesn't count; the Archbishop of Lisbon was only given the title of 'patriarch' in the 18th century).
Short version, while the status of Constantinople as a patriarchate of the Pentarchy post-dates the firm establishment of a recognised Bishop of Rome, the establishment of monarchical episcopacy in Rome almost certainly post-dates its establishment in Antioch and Alexandria.
Finally, and I mean this in the nicest possible way, I don't know where you found that chart you shared of the supposed descent of various Christian denominations, but it was astonishingly poor from a historical perspective. I respectfully submit that your grasp of early church history isn't perhaps that strong. I can recommend some books written by historians of the early church who are writing from a theologically neutral perspective - rather than from an inherently Catholic or Orthodox perspective - if that would help.
