First of all the Eastern Orthodox Church is not false as such but merely schismatic from the Catholic Church. The main reason is that the Bishop of Rome was always the head of the Catholic Church and always had the power in Church law to remove and replace Eastern Patriarchs (though this ability was often blocked by the Byzantine Emperor) and thus breaking from the Pope was splitting from the Catholic Church (the Bishop of Rome described as the source of sacerdotal union by St Cyprian of Carthage in the middle of the 3rd century for example). Also the same Eastern Patriarchs that split from the Catholic Church were quite often in heresy historically speaking and had to be brought back by the Pope and also they were quite often puppets of the Byzantine Emperor.
As for Councils being the deciding factor mostly it was a Council in union with the Pope as many councils of bishops (who were always the people with a vote in councils, it was never a popular vote) backed things like Arianism/Monothelitism etc. but weren’t ratified by the Pope. In fact the Patriarch of Constantinople usurped a lot of power outside of his established boundaries of Asia, Pontus and Thrace by claiming authority over the whole Balkans, the Patriarchs of Alexandria, Antioch and Jerusalem and even southern Italy and Sicily and calling himself equal to the Pope and Ecumenical Patriarch, this all being given to him by the Emperor not Council.
Papal Infallibility was voted for by a Council, albeit a more recent one, though there were early mentions of Papal Infallibility and pronouncements made, even in the third century, possibly earlier, though admittedly it was less specific.
While the Eastern Orthodox have undeniably kept Church practice the same more than the Catholic Church since the split they have changed their views on doctrine and morality by stopping believing in Original Sin and Purgatory (both voted for in the Synod of Jerusalem) and allowing contraception. Furthermore they insist everyone uses the Byzantine Rite which is ridiculous as it is only one of the original Rites still uses in the Catholic Church and more recently created than the Latin, Syrian and Coptic Rites.
Also unlike the Catholic Church (with the Pope) they have nothing unique to say they are the one true Church, as opposed to the Catholic, Oriental Orthodox or Assyrian Churches, so that claim doesn’t make as much sense, whereas a claim to be part of the true Church would