The problem with the “First Church” claim is that there are many denominations which claim that they are the only one and true “first” church:
the Orthodox,
the Ethiopians,
the Miaphysites,
the Chalcedonians,
the Catholics,
etc.
Secondly, it is acknowledged that Rome had a seat of pre-eminence amongst the Early Church just in the sense of first among equals. Akin to how the Archbishop of Canterbury has primacy among Anglicans. It never had the power to make doctrine, or to tell other Churches what to believe, and Paul even proves in the Epistles how he is equal to Peter in all things. If we take Scripture seriously, Paul is equal to Peter among the Apostles, and there is no supremacy of Peter over Paul.
Here then is the problem with the claims of Catholics – the granting of the “first among equals” alone. In addition to this there are many doctrines they’ve introduced later which go contrary to the doctrines that the Early Church had stood for:
-there was no papal infallibility, even among Catholics, until 800s or so
-the “rock” of Peter was interpreted to be his faith, not his person, until 700s or so
-there was no monasticism until the 500s
-there was reverence for celibacy until the 500s
-there was no Mariology, with the first unofficial reference to her appearing in the 300s or so (and being absent from the Nicene Creed altogether, as irrelevant).
These were all the things that were introduced by the Catholic Church over time, especially papal supremacy introduced during the Dark Ages over an ignorant Europe, to extend the Pope’s power. For these reasons, it is argued that Catholicism has quite clearly left the Early Church model, and is no longer a representative of original Christianity.