Sorry, I was going on this, that you said.
"I believe that the mystery remains because God wants our faith.
God could prove that He exists if He chose to, so the fact that God has never done that must mean that if God exists God wants our faith."
Is there evidence or not?
Perhaps you are confusing evidence with proof. There is plenty of evidence that God exists. Even this conversation is evidence of it. But evidence is not proof. So if one chooses to trust what evidence they have (it will be different for everyone) even though they can't know it to be so ... that's an act of faith.
If they choose to presume that God exists based on whatever evidence they feel they have, then that is their "belief".
If you, on the other hand, are looking for
proof of Gods existence, you won't find it. Because there is no possible way for a finite human being to validate the existence (whatever that would even mean) of an infinite being. I will give you an example.
Imagine that 'God' is hovering in the air in front of you right this minute in a "blaze of glory", and is telling you that He/it is God. How could you possibly determine that this was actually God? As opposed to perhaps being some very clever magician's trick. Or the deceit of some demon? Or perhaps it's some advanced alien being from outer space appearing to you in the way it thinks you will understand? Or perhaps it's some sort of anomaly happening in your brain?
The point is there is no way that we can verify the actual existence of what we call "God" because what we call God is a concept that is beyond our ability to comprehend or verify. It is infinite while we are not. It creates and transcends even existence, itself.
The complaint that there is "no evidence" is false. It's based on the errant presumption that "evidence = proof (that I can understand)". There is lots of evidence. But it does not rise to the level of proof for two reasons. One is that the "decider" of what he will accept as proof doesn't want it to. And therefor won't allow it. And second, because we humans don't possess the capability of recognizing what proof of God would even look like. And certainly not that we could verify.
So the choices left to us are faith, belief, or doubt.