What evidence do you have that you exist? You are asking the impossible. You are asking me a question like "prove to me that God exists and is exactly as He appears in the Bible. I can give you reasons I believe, but no proof that will satisfy you.
Nevertheless, I'll take a crack at it. I believe what I believe for the same reasons you believe what you believe. I accept by faith that the Holy Spirit led the early church concerning what was canonical. My faith is bolstered when I actually study the books and see the unity within them. They have a harmony that the other "books" do not have. And, I'm not talking about just the New Testament. The New Testament is in harmony with the old. The old with the new. They tell one story from the beginning of time until now of One God and His message to mankind. No other group of writings (be they the pseudo this and that or even the apocrypha) comes close.
So, frankly it wasn't the Church's judgment I trusted in to get the Canon right. It was the Holy Spirit's guidance. He can make an ass speak. He can certainly make a group of people from disparate backgrounds do what He wills them to do.
Regarding the church's judgment, I will take Paul's que here.
ACTS 17
10And the brethren immediately sent away Paul and Silas by night unto Berea: who coming thither went into the synagogue of the Jews.
11These were more noble than those in Thessalonica, in that they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so.
The "Church" isn't wrong on everything. But what they make pronouncements upon MUST stand up to the Scripture test. If it contradicts Scripture, it should be tossed. That is the standard. Subjective? At times. But not so much as you "non-sola-scriptura types" would imagine. There is one Spirit. He leads me in the right path.
I will agree with the assertion that 'what contradicts Scripture' must be tossed, however, with two provisos: first, provided the Scriptures are 'rightly divided', that is rightly interpreted, and second, that we accept the judgement of the Church, not the Christ-denying rabbis who met a Jamnia in 90 AD, as to what constitutes the Scriptures of the Old Covenant.
However, I have observed that most 'sola scriptura' types go a step further than what you have advocated, and toss not only what contradicts Scripture, but anything their discursive human reason cannot derive from Scripture. 'Sola scriptura' invariably disolves into private interpretation, with a hermeneutic that ignores the problem of intepretation, and rejects the context in which the Scriptures were written and canonized--the life of the Church--and insists that whatever the interpeter cannot *prove* from Scripture is false. This last is a much stronger assertion than you made, and is contrary to the plain word of Scripture, since a report of one of the deeds of Christ alluded to by the conclusion of the Gospel of John could not be proved from Scripture, but would not be false.
Christ promised His disciples that the Holy Spirit would lead them into all truth. It seems to me a very odd faith to believe that that leading into all truth is confined to the fixing of the canon of Scripture, and does not also extend to the Church judging the content of the Proto-Evangelium of James of the Didache, of accounts of the lives of the saints (as, for example The Martyrdom of Polycarp) to be sound and truthful, even though none of these documents meet the standard of undoubtable Apostolic authorship. (The Didache is a set of 'lecture notes' on the teachings of the Apostles, refered to in ancient patristic texts, a copy of which was found in the library of the Great Chruch of Constantinople in the 19th century.)