Is having true predictions an end in itself?
Only if you care about what's true. If you don't care about what's true, you have no reason to care about science. Or about your faith, for that matter. (If you adhere to a Christian faith, your pastor won't like it if you tell her "I don't care if it's true that the trinity exists" --- and rightly so.)
We may have methods to accurately answer most questions we ask, but how do we decide which questions to ask?
That's an independent question. Without a pre-existing proposition you seek evidence about, or faith in, neither has anything to attach to.
The more I think about your initial post, the less I believe I know what you're getting at. On the face of it, it seems hard to discuss your topic without circular reasoning. Evidence, by definition, is a thing or a set of things that furnishes belief. So why do we believe in evidence? Because it wouldn't be evidence if we didn't. Take believers who pray to their deities for the answer to important questions in their life. When they hear an inner voice giving them the answer, that's genuine evidence to them. In their mind, their faith in the answer doesn't rest on an alternative
to evidence; it rests on evidence.
Against this background, the question "why do we believe in evidence?" seems akin to asking, "why are liquids wet?". The sentence sounds like a valid question because that's what it is syntactically. But semantically, it's really not. It's just philosophical-sounding nonsense. Can you re-state what you're getting at in a form that doesn't involve this kind of nonsense?