I remember hearing stories of Jesus healing lepers and epileptics (people possessed with demons), or reading about Joseph Smith healing malaria afflicted Nauvoo saints. I remember being skeptical of these stories, just like I was skeptical of the anecdotal testimony-builder stories I heard over the years. I also remember reading stories about Joseph Smith trying unsuccessfully to heal fellow Zions Camp members who were sick with cholera (fourteen of which later died). Why could Joseph (apparently) heal malaria, but not cholera?
It seemed that Mormons today also know that Priesthood blessings don't really work; at least that's the way they act. Why else do LDS men usually only give blessings to people suffering bothersome cold and flu symptoms, or other minor illnesses that normally resolve on their own? Why wait a few days into the illness to give blessings - right at about the time the illness starts to resolve anyway? David Hume wrote about this sort of miracle, stating that "Nothing is esteemed a miracle, if it ever happen in the common course of nature." Minor illnesses naturally resolve because our immune system has evolved to eradicate them, not because a miracle has happened.
However, when illnesses are serious (cancer, heart-attack, stroke, chopped off limb, etc) Mormons get to the hospital fast, just like anybody else. We trust modern medicine and all the science and technology behind it to heal us - not magic priesthood blessings. Again, Mormons act like they don't really think priesthood blessings can heal people. Otherwise, we'd admister priesthood blessings first, and rush to the hospital second.
Also, people will pay attention to the times when their illness gets better, and chalk it up to God's healing power. However, they will quietly forget about all the other times when prayers and priesthood blessings didn't work. This is a common blind spot of human psychology called confirmation bias. Also, if prayers and priesthood blessings aren't answered, we can always blame it on our lack of faith, or perhaps it wasn't God's will that we should be healed. Maybe God is just trying to test our faith?
However, faith healing has been shown to be ineffectual time after time in carefully controlled studies of the effectiveness of intercessory prayer. It just doesn't work - even though people like to think that it does.
Even if you believe in God, I think it's pretty arrogant to think you can control God's actions by performing simple human rituals. Why do we think God will intervene for us, but not all the kids dying of malaria and AIDS in Africa? Why do we think it's fair for God to stop the course of nature for us, but not for others? Because we happen to put olive oil on someone and say a prayer with out hands on their head!?
Deep down, I don't think a lot of Mormons really believe priesthood blessings work either. I'm not sure I did, even though I professed to. Again, our beliefs are revealed by our actions. If we really believed in faith-healing, then we'd act like those Christian Scientists, Jehovah Witnesses, Scientologists, and Amish, who eschew modern medicine, and put their complete faith in God when they get sick. They are the true believers (despite the fact that they sometimes let their children die of easily manageable diseases).
I'll end on a very personal note: My wife's sister died 9 years ago of cancer. She had a young baby and new husband at the time. They were in the prime of their young lives when a rare form of cancer was diagnosed. My wife's family, and just about every Mormon who knew them, were praying for her complete recovery. Specific priesthood blessings had been given to her that stated she would be healed.
She wasn't. She died despite all the prayers, the fasting, the blessings, and modern medicine. It was surely a blow to everyone to lose her. I struggled for a long time wondering why God didn't heal her. We had faith. She was righteous and faithful. She had priesthood blessings given to her promising her recovery. She had access to the best modern medicine had to offer. People were praying for her. And yet, the desired result wasn't achieved. The only other excuse left was that it was God's will that she be healed. I hated hearing people, usually her own family, say that. I thought it was made no sense. Why would God need her more than her own child? Why WOULDN'T it be God's will?
I came to the conclusion that God just wasn't as involved with our lives as I had previously thought. It was my first move away from a personal God who was aware of us, and who intervened in our lives - towards a God who let nature take its course, despite our efforts to control Him. This was the God of the Deists, who, although more consistent with nature than the interventional God of monotheism, was just not that satisfying to worship.