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Well, I imagine it's because of the 'supernatural' aspect of such an occurrence. Sure, amazingly good luck can have some profound results, but I've never personally heard of healing or prophecing. As for random power and 'magic gravity', it would still fall under supernatural since it's not something found in nature.
Indeed, however, a "higher power" and the "supernatural" isn't the same in that way. The supernatural might be a mindless force, like, uhm, The Force, without it having to be what we would call a "God".
Yet, I almost always see "miracles" being tributes to a higher power, without any thought for that there might be other explanations. That's why I don't think they are very good evidence, as they are being biased from the start.
Anyways, I shall continue my arguments.
Arguments from holy scripture
Seeing that holy scripture is part of most religions, and therefore also tightly tied to the idea of a higher power (God), I'm going to make a few arguments about them.
A) A lot of proof from holy scripture uses circular reasoning
One thing that a lot of people seem to miss is, that you can't use something to prove itself. If I was a murder suspect, you can't use my testimony to clear my own name.
However, I've more than once seen miracles in holy scripture, that are first mentioned in the scripture, and then proven in the scripture.
One example of this is the resurrection of Jesus, which is first claimed by the Bible, and may Christians (I've seen a lot of articles call it a "historical event") think it's proven because the Bible says that there were several hundred eye witnesses. But this would be the case of circular reasoning. How can we verify those witnesses? The same goes for a lot of miracles in the Quran and so on.
What you would need is some outer source that can confirm claims.
B) Proving part of something won't verify everything.
Book of Drakim:
1. The sun is made of hydrogen
2. The sea is salty
3. The moon is made of cheese
Now, we know that 1 and 2 is right. We can verify that those two points are correct based on our knowledge. However, does that make 3 correct too? Of-course not. It would be quite foolish of us to believe 3 simply because 1 and 2 is correct.
However, this is something I've seen a thousand times when it comes to holy scripture. Let's say that the resurrection of Jesus was proven. Would that, however, prove that there was a global flood? Would it prove that Adam and Eve existed? Sadly, many seem to think so, even when we can clearly see the error by putting it as an example (book of Drakim).
Personal Testimony
While a personal encounter with something anomaly, such as a ghost may be very convincing for yourself, it won't hold up if you want to prove the existence of ghosts to others.
It may sound cold, but humans are simply too easily tricked. Even perfectly normal humans can be convinced of something fundamentally wrong by a very powerful experience.
And the fact remains that we dream. We day-dream. We even hallucinate. We can have the most convincing vision ever, about just about anything. We aren't perfect machines, but humans, and humans can often be mistaken. Our memory changes constantly, often by nothing but time or suggestion.
I'm not saying that human experience are useless. We can still get a murderer conviced of murder based on the accounts of eye witnesses.
However, as I pointed out before, the claim of a higher power is quite a spectacular and reality shaping one. It requires more than something that may be a dream.
