No matter what form it takes, religion is the intuitive discernment of the Infinite. It seems to me the incessant demand for evidence for something that, by definition, is infinite, shows an inability to intuit the presence of something which, by virtue of it being infinite, more than a thing.
Some years ago biorhythms were all the rage: you could go to any store and buy a book that charted good and bad days. I was interested, but basically thought it was a bunch of hooey. Now, years later, Ive monitored the inner workings of my life to know that while the charts and graphs were a great way for people to make money, life does move to a rhythm that no clock can accommodate. There is absolutely no way to provide evidence that I am aware of non-physical forces at work. A light touch on the skin is more subtle than a punch in the face, but that doesnt mean the former is supernatural. A black eye is evidence that someone punched me in the face, but how can I show that I was lightly touched?
I know that many religionists believe in God in a way that makes their religion vulnerable to criticism, and most of those believe as they do because it was something passed down to them rather than something experienced. Nevertheless, there are those that, while operating from obsolete ideas, genuinely intuit the presence of something working through them. Their concept of God may be underdeveloped and critics have every right to question the concept of God they employed, but there is no reason to ask for evidence because no evidence is possible other than the life of the religionist.
I think what you're referring to is a believer's internal, subjective experience, correct? And spiritual intuition? I think there are a few problems with these though.
One is that, by coincidence, people just happen to have spiritual intuitions that match their cultural upbringing. Catholics have visions of the Virgin Mary, animists experience connections with vital spirits, and Evangelists talk to Jesus. Another is that doctors can literally induce vivid religious visions by stimulating a person's brain. In fact, there is a kind of epilepsy that causes these visions. But more important to me is the basic idea that our brains just work a certain way, and we're getting where we're starting to know what that way is. One is that we think in terms of actors and agency, of intentions and stories, even when there isn't one. That's just how we evolved. We are pattern seeking creatures, and we will see patterns when there isn't one. We can't help it. So, due to our normal brain functioning, we are prone to erroneous spiritual intuitions.
That doesn't mean there isn't such a thing as a real one, but more that there would be no way of knowing a true one from a false one.
It's possible that by insisting on evidence, we are missing out on some important truths. However, by letting go of evidence, you are at risk of believing things that are false. Evidence = risk of false negative. No evidence = risk of false positive.
So I'll take my chances with the evidence.
btw, I really think we all go the evidence route in our everyday lives, and let it go only when it comes to spirituality, religion and the supernatural. Or at least, most of us. I don't think many people say, "My doctor told me to take anti-biotics, but I'm disregarding the evidence and going with my spiritual intuition and throwing them away." Or even, "My brakes are squealing, but my heart tells me not to take them into the mechanic."