It's not that it is independent of evidence. It is that the evidence is not sufficient for proof.
For example, the apparent beauty and organization of the universe intuitively implies a designer. If we found a computer on Mars, we would conclude it was created by an intelligent being because it implies design. This implication of design can be very compelling evidence for a great many people. But it is not sufficient to be overwhelming proof. For example, the entire idea of beauty could simply be an accidental by-product of our evolution and we may be seeing design where non exists, much in the same way we see faces where none exist.
I should have stated it more succinctly. It is a belief based insufficient good evidence. There is "evidence" for unicorns as well....it's just not sufficient or good enough to warrant belief.
Beauty is a human construct. Nothing in the universe is inherently beautiful or ugly. That is a quality assigned by the observer. So the things we might describe as beautiful are only so because we say they are so.
Also, If the beauty and organization we see in the universe is evidence for a god, then the disorganization and ugliness we find in the universe must be taken as evidence for the absence of that god. And there is both.
Your computer analogy fails right out of the gate. It is just the old watchmaker argument from hundreds of years ago.
We know the computer has a designer because we are the designers. we can find blueprints, wiring diagrams, factoriis, we can talk to the workers in those factories. So we have knowledge of how it is made.
The argument also fails because you are differentiating it from everything around it and saying it stands out because it was obviously created by an intelligence. Well, why didn't you choose the rock it was sitting on instead??? If everything was created intelligently, how were you able to single out the computer???