We are not forced to choose between all or nothing. Let me mention some evidence that supports biblical authority. First, Scripture itself claims to be inspired by God. Writers such as Paul claimed to write with authority derived from his commission from God. This is a faith claim, not a proof.
Tradition supports biblical authority.
Christians throughout the centuries have found these writings to be useful and reliable, for both faith and practice. These books tell us what sort of God we believe in, what he did for us, and how we should respond. Since God went to the extreme of revealing himself in flesh, we believe that he would also ensure that the revelation be faithfully preserved. This cannot be proved, but is based on our understanding of what God is doing with us. Of course, the fact that a moral authority is needed does not prove that the Bible is that authority, but history at least shows that the Bible was useful in reforming the problems, and its usefulness came because people accepted it as an authority.