I think you have a point about auto suggestion, time, improved diet and other such assumptions as to why some prayer may be answered without a need for God's intervention. Hope and expection actually aid the immune system. I would disagree that God has somehow been eliminated from the process. We would have to look at claims of healing such as cancer, broken bones and the like to see if there are any proven cases where healing has taken place in such a way that no natural explanation can possibly be given. Such documented cases abound and may explain why 3 out of 4 doctors believe in at least some cases of supernatural healing in the world today. Most of the arguments against healing that I've seen in this forum are highly emotional and speculative and seem determined to prove the belief in methodological naturalism.
Yes, I studied hypnosis deeply and I can assure some diseases can be cured with the power of the mind.
Most miracles you see in the church are just subtle suggestion: random pain disappears, person stops smoking, person stops drinking alcohol, person gets a better outlook in life and stops depression.
ALL these can be done in the church. None of these is supernatural. A bad mindset CAN cause diseases, even cancer.
I think this thread is probably the best argument for faith healing.
Amputees will never get healed though. It's a PHYSICAL problem. An arm won't grow back. Heck, not even hair in bald men will grow back.