For a short time in my teens and early 20s my family were Jehovah’s Witnesses.
They take pride in declaring themselves a Bible-based religion. In fact they make note that they reject any and all Tradition that came before. For them Jesus was not God’s revelation to man. Only the Bible in written form was God’s ultimate revelation to man, a written work that would await the coming of the last days to be understood by their leaders. “We believe in the Bible, and nothing but,” is what they would make us say when we knocked on people’s doors.
Because they believe in understanding the Bible in a vacuum, void of and rejecting the rich heritage and Tradition that formed it, they end up with such unorthodox beliefs such as no Trinity and soul-sleep.
So you can have a “Bible-only” based religion or church. But as you said, it causes a great paradox.
The Bible is actually based on a religion. It was not meant to be used or understood in a vacuum or independent of that faith. That is why Catholics learn so much about the Jewish faith and have many aspects that are similar.
Having the Bible but rejecting the religion that produced it is like taking the writings of the Dalai Lama and following his writings but rejecting him and Tibetan Buddhism. Sounds silly doesn’t it? Why do it?
If the writings are a reflection of the religion but you reject the religion from which they came, you are left only with a reflection.
And if you attempt to read and follow a reflection you will only end up getting things backwards.