Studies on intercessory prayer (where God makes no difference)

Someone asked, "[person x] pointed out the study that patients who know they’re being prayed for have less of a chance to be healed. That’s what I would expect if God didn’t exist. How do you address the problem of unanswered prayers?”

I responded:
a) My personal suspicion is that God normally does not answer prayer with miracles, but only occasionally and in special circumstances. Prayer is more about the obedient Christian connecting with God as God, and submitting our fate to him. Miracles happen, but God will have special reasons for doing it and tends more to work through the hearts of believers. He has reasons for permitting evil (see BeliefMap) and disallowing us from always praying it away.
b) Moreover, there is something sacreligious about treating God as less of a person and more of as a dumb animal we can catch with our nets. If you consider God's personality in the Old Testament, I don't see God allowing morally disinterested scientists from "catching" him and making him jump through hoops, all the while having no interest in God Himself as God.
c) Not that I believe this, but here's an option for someone who dislikes A and B. At least plausibly, God wants the evidence to be a little harder to find, to allow for people who want to delude themselves. A prayer study confirming Christian prayer might be too overt an evidence for God's purposes (see the Divine Hiddenness section). After all, it would ostensibly transform the whole planet in almost coercive ways towards Christian theism. That might result in more theists, but fewer people seeking God for good purposes and genuinely loving God in the long term.
d) Fwiw, that prayer study is kind of silly. It was entirely or primarily for Catholics, as I recall, and the participants were to only read a short artificial prayer from a card!