Please describe to me what the process for a sceptic applying their own beliefs to the scrutiny you are talking about. I am really not understanding what you are wondering.
The same process you would follow if someone came up to you and claimed to be Amelia Earhart. You would first doubt, then question, then examine whatever evidence is presented.
It seems to us atheist skeptics that theist skeptics are willing to apply this procedure to everything other than their own religion's claims to gods or miracles.
Here's an example, from my own experience. While I was attending my Pentacostal Christian church, there was what I suppose you'd call a revival meeting. A bloke was brought in from outside, and he proceeded to "heal" people. One of the people he "healed" was an elderly woman in a wheelchair. After his prayer and his laying on of hands, she stood up out of her wheelchair and walked unsupported.
"A miracle," I cried. "Hallelujah". I did not question the miracle that Jesus had laid his hands on this woman and healed her of her lameness. This was talked about in the church for some time afterwards.
It was only years later that I realised that
I knew nothing whatsoever about that woman! I had never met her before. I didn't know
why she was in a wheelchair, I didn't know
how long she'd been in a wheelchair, I didn't know whether
she got back into her wheelchair after the meeting!
A few possible explanations came unbidden into my mind.
She could have been genuinely healed of whatever infirmity or disability she was suffering from.
She could have been swept up in the moment and overcome her disability due to adrenaline. In this case, she would not have been "healed" at all, and would have got straight back into the chair after the meeting.
She could have been someone who didn't normally use a wheelchair at all, but was simply offered one as a courtesy of the organisers of the meeting. In which case she wouldn't have been "healed" at all, because there was nothing to heal!
I have since heard that some revivalists will bring wheelchairs to the meetings and offer them to the frail and the elderly
just so that they can get them out of the chair and claim that it was a miracle!
I think this is a far more likely explanation of something that I had previously been
absolutely convinced was a genuine demonstration of the power of the Holy Ghost. I don't know whether the revivalists in question engaged in active deception. I have no way of knowing. It was a long time ago. But it's one of the options.
It seems to me that theist skeptics
do not see the options where their own religious convictions are concerned. I and others wonder why this is so, especially since theist skeptics are as ready as we are to doubt the claims of dowsers, psychics, chiropractors and remote viewers. To me, claiming a miracle is
of the same order as claiming psychic power. To theist skeptics, it isn't. And I wonder why that is.