I often like to point out that Catholicism needs to deny the Sufficiency of Scripture, claiming for itself authority to interpret it, not because this view is correct, but because Catholicism does not fit very well with scripture at all. If you had never been exposed to the Catholic faith, and had nothing but the Bible to teach you theology, you'd never come up with most, if any, of Catholicism's unique doctrines. You'd certainly never conclude that the Catholic Church is the one true Church established by Christ, and it would be ironic if you did, since you would then be privately interpreting the Bible to mean you cannot privately interpret the Bible.
But of course, just as many ex-Catholics will testify that they left the Catholic Church after a diligent Bible study, so also will many Catholics testify that it was their study of the Bible that lead them to their Catholic faith. Problem 1 with this claim is that they follow this up with a very unbiblical case. They don't like to talk about scripture, they like to talk about (their version of) history. They don't cite scripture even half as much as they cite the Church "Fathers". They spend more time repeating the lie that the Catholic Church produced the Bible than actually relying on what's inside. When a "Protestant" cites scripture, suddenly "you don't have the authority to interpret that book", and on the rare occasions when they cite scripture, it is in the same eisegetical way as every pseudo-Christian religion in the world; meaning is read into a verse, rather than a contextual reading being used to understand what it says.
If this is the route they lead others down, it can be reasonably assumed it is also the route they travelled. Thus, their claim to have converted based on the Bible can be dismissed. Let us think about it this way: Could you actually find out about Catholicism's unique doctrines from scripture? Could you find a Papacy, the Marian dogmas, Purgatory, the 7 Sacraments etc. in scripture? You can apparently see them now that you have been exposed to Catholic theology, but without the Catholic Church being in some way present, you would not even know these doctrines existed.
The same can be said for a number of "Protestant" doctrines. This is simply because of the ambiguity in the term "Protestant". The term "Protestant" is, by its very nature, a negative statement, illustrating what we don't believe rather than what we do. In effect, "Protestant" simply means "non-Catholic". For that reason, some Catholics actually classify other cults, such as Mormonism, as "Protestant".
And this is why you will so rarely see me use the term "Protestant" without air quotes. Due to its common acceptance, I cannot completely avoid using it, but I do not like to use it as a self-descriptor, for the same reason I think it's daft to call me an atheist because I reject Allah.
Personally, I don't like to play the denominations game at all. They're divisive, they're redundant, they're subject to change, they're subject to error, they're just not for me. For that reason, I accept two kinds of titles.
The first is specific titles accurately describing a belief I already hold. "Biblical Creationist", for example, tells you that I hold to the account of origins as found in the Bible, primarily in Genesis. This is a specific, and more importantly positive title. It identifies what I do believe, only identifying my unbelief in other views by logical extension.
The second is simply "Christian". This is what the faith has been called from the earliest of days, even being mentioned by name 3 times in scripture. This actually includes the first use of the term (Acts 11:26). This term is a summary of the faith as a whole, and so all who believe should accept it. Just as Biblical Creationist is descriptive of what one does believe, so also is Christian descriptive of what one at least strives to believe.
Just as "Biblical Creationist" negates belief in Evolution, "Christian" does, or at the very least should, negate Catholicism. This isn't because there are no Christians caught up in the Catholic Church, but because the Catholic Church itself is not compatible with Christianity. From the minor to the major, there are conflicts between scripture and Catholic Doctrine.
For this reason, Catholicism is actually not relevant to my faith in any special way. Personally, when I became a Christian, I had no clue what the Catholic Church taught, and I was spiritually immature enough to believe anything, or anyone, claiming to be Christian actually was. But I didn't stay immature. I studied, I grew, I developed my theology, and I'd read the Bible around 3-4 times cover to cover before I really started to look into what Catholicism teaches. Keeping in mind "Protestant" just means "against the Catholic Church", I became "Protestant" without changing a single view.
To this day, not a single view I hold is directly affected by Catholicism. In fact, I could easily preach against Catholicism without directly referencing it a single time. Of course, in one sense, I don't need to. Scripture does claim to have the purpose of reproof and correction (2 Timothy 3:16), and so it is both permissible and logical to say "Catholicism teaches this, but the Bible teaches against that in this way". But at the same time, because Christianity is based on scripture, a faithful exegesis of scripture will naturally oppose Catholicism without ever once mentioning it.
We could start off with arguably Catholicism's smallest lie: The perpetual virginity of Mary. The simplest and easiest way to refute this lie without direct reference to it would be to simply preach on her life. In doing so, we find that while she was a virgin when Christ was conceived, as was foretold, she also married Joseph, and Matthew tells us he kept her a virgin until Jesus was born, but after that, she did have other children. She is even often found walking around with them, and Jesus makes a point about this. If we wanted to get really deep, we could even bring up Messianic prophecy, such as Psalm 69, when the Messiah's "mother's sons" reject Him. With all of this included in a single sermon about Mary, you can refute the perpetual virginity of Mary without even referencing it, and the onus is on the Catholic to protest.
But no one goes to Hell for believing Mary deprived Joseph of his marital rights. People do, however, forfeit salvation by adding works to it. Could we as easily refute the Catholic view of salvation without direct reference to it? Certainly, first of all because Catholicism is not the only perverted gospel. There are many other faiths that teach salvation must be earned and/or maintained by works. Even some "Protestant" denominations still teach that baptism is essential for salvation. The Catholic Church does indeed teach that works are necessary for salvation, and so combating those "gospels" is directly combating Catholicism, even without mentioning it.
But you don't need to attack false gospels to present the true one. Indeed, the Gospel summary on this very ministry contains but one vague reference to false "gospels" at the very beginning. In this summary, I do not address Catholicism a single time, I simply present the truth as it is found in the Bible.
And that there really is the point. Christianity, unlike "Protestantism", is not the non-belief in a single false Church, it is a set of beliefs that just happen to conflict with this false Church. A Christian who becomes a "Protestant" becomes a "Protestant" not by accepting whole new doctrines, but because the actual new doctrines have presented themselves, and he opposes them. But these doctrines, while they are "old" now, are absent from, and usually contrary to, scripture. Thus, it is only logical that a Christian will reject them when they show up in their own lives.
And so the long and short of that is that actually, Christians need neither be Catholics nor "Protestants". The Catholic Church does not have the relevance it pines for, neither being the default denomination, nor meriting a term for rejection. For this reason, I do reject the Catholic Church, but I do not accept a special name for it, nor adapt my theology accordingly. Most people, studying scripture, will come to the same conclusions.