|The Catholic Church can trace it's roots and history back to Peter the Apostle.|
According to the law of identity, a thing is defined by its attributes. The written teachings of Christ and the apostles (ie. the New Testament, which all conservative Christians, Protestant or Catholic, would at least agree would not contradict their oral teachings) do not contain many doctrines which are today defining attributes of Catholicism, such as Mary-related doctrines, doctrines about the Pope and priesthood, etc. Indeed, I believe that the New Testament directly contradicts much Catholic teaching, and that Peter and co. would have been considered heretics by modern Catholics and vice versa. Catholics would obviously disagree with me there, but IIRC they do hold to the "development of doctrine" belief, which is an admission that "Catholics" like Peter did not believe many of the same doctrines, either at all or to the same extent, that Catholics today believe.
Given that, "Jesus founded the Catholic Church" must at most be read as "Jesus founded a Church which was markedly different in doctrine and practice from the Catholic Church today, but which turned into it over time". Referring to Jesus, Peter or any of the early Christians as "Catholic" is therefore misleading without that caveat.
Then you have to ask what is meant by "the early church developed into the Catholic Church". After all, I could equally claim that the early church turned into the Reformed Baptist church - it did, in the sense that there were always believers who believed a variety of things, some of which at some periods of history corresponded with what later became popularly known as the Reformed Baptist faith. In that sense, I suppose there's continuity of sorts. (But Reformed Baptists don't tend to care when a doctrine was developed/formalized/popularized, if it is clear it was originally taught in Scripture - being twisted or lost for 1000 years doesn't make a teaching any less true or less "founded by Christ".)
Similarly, the Early Fathers believed all sorts of things, more or less heretical, some of which are very modern-Catholic in flavour and some of which are downright Protestant - and many of which contradict each other, the Early Fathers being by no means a unified front. Picking out certain doctrines and certain quotes from certain members of the early church and labelling them "Catholic", with the rest discarded as heretics, can provide the illusion of continuity and unification which history does not support. It's revisionist and selective.
Ultimately much of the division between Protestantism and Catholicism comes down to authority and truth. Protestants believe that, say, the early Church was authoritative in the matter of ratifying the canon inasmuch as it was true; Catholics believe that we can know the canon is correct because the early Church had authority. Protestants believe that the doctrine of, say, sola gratia is authoritative because it is true, as determined from Scripture; Catholics believe that the doctrine of, say, the the immaculate conception is true because it is taught by the Church, which is authoritative. So a lot of dialogue between Protestants and Catholics ends up missing the mark: saying, sans Church authority, "The Bible says X" to a Catholic is an unconvincing to him as saying to a Protestant "The Catholic Church teaches X" is unconvincing to him. Different presuppositions.
So I'd suggest the really important question to ask yourself, if wavering between Protestantism and Catholicism, is "which presuppositions do I believe, and why?" Do you believe that the Bible can be understood by people like you, or that it requires a Magesterium to interpret it; and why? Do you believe that Jesus gave infallible teaching authority to members of a specific denomination, or not; and why? Do you believe the teachings of the Church today are in accord with Scripture or not; and why? Do you believe the Pope has authority/sits in the seat of Peter, or not; and why?