Catholics and Orthodox note a different point.
Tradition preceded scripture. Sticking strictly to the New Testament, (because the dates are easier to manage in discussion, but the same situation applies to the Hebrew Scriptures), there was no scripture, at all, prior to the fifties, when Paul began writing his epistles to various communities. Even when he wrote his letters, they were hardly recognized throughout every Christian community as “scripture.” First they had to be copied and sent out from the original locations to which they were sent. Then, time had to pass in which they were collected together. None of the canonical Gospels could have been in circulation prior to the seventies, (or even much later). The same impediments of not having been written and not being distributed to all Christians applies to all the other writings, as well.
Thus, there could not have been a Christian Scripture for a minimum of twenty years follow the departure of Jesus. Looking at the historical record, it actually seems that the gathering of such writings by Marcion just before 150, (in which he overtly excluded any writing associated with the Jewish people), was the first formal effort to create a canon, so we are really talking about a church that did not have a New Testament for well over 100 years.
During that period, it was the tradition of the teachings that informed Christianity. In fact, the standard by which various works were admitted to the canon was based on their adherence to the understanding of tradition held by the church in that period.
There are any number of objections one might raise concerning the selection, inclusion, and exclusion of various works as declared by the Catholic or Orthodox leaders. It is also possible to make the argument that only those portions of tradition that actually got recorded in what we now call scripture are valid. However, one cannot legitimately claim that the church has always had scripture to preach and explain the words of Jesus. For a significant period of time, the church grew and developed without any such scripture even existing. It is the Catholic and Orthodox position that they are maintaining the tradition that has always been a part of the church. If tradition is contradicted by scripture, it cannot be acceptable, but those wrinkles were ironed out long ago and from the perspective of the Catholics and Orthodox, there are no longer any contradictions. (On some points, obviously, various Protestants and others disagree, but the Catholics and Orthodox would tend to say that the interpretation (of either scripture or tradition) of the objectors is in error.)
As to tradition not being infallible, we could turn that around and note that scripture, arising from fallible tradition, is also fallible. That would hardly be a Catholic or Orthodox position, but it is the natural conclusion of rejecting tradition as fallible.