The influence of religious lifestyle and prayer in statistics

Why wouldn't Christianity and Islam have an impact? After all, these are religions that believe in one God.
As I wrote, one can rationalize this effect in all sorts of natural ways or suspect the research method. Therefore, I do not rely on them to *prove* the power of prayer. But I can use them to *disprove* the opposite claim that there is no statistical difference between religious and non-religious people. The heretic may not be convinced, but the believer has no reason to doubt his belief on the subject.
Personal feeling in everyday life is not a relevant indication in this context. There are many religious people who do personally feel that their prayers are answered. And I would not be quick to make assumptions about such and such foolish followers without checking them, just as those who assumed that there would be no difference between religious and secular people were wrong.

Christianity, at least in certain shades, is a heresy, and in all shades it is an obscene error. So if God helps the worshippers of a heresy and the erring, then He can also help atheists. You are losing the ground from under your claim.
I disagree with your starting point. I argue that the Sages and all our predecessors also had no basis for assuming the divine influence on life. Therefore, I do not give much credit to this belief in advance, and I do not think that it is held to be true until proven otherwise. On the contrary, for me the holding is that there is no such influence until proven otherwise.
Anyone who wants to can hold on to all sorts of things that can be interpreted in all sorts of ways. Personally, I don't see any indication of such a difference and such an influence.
There is no difference between using studies to prove belief and using them to reject refutation. Either you believe them or you don't. If you don't believe them then it is correct to say that there is no indication of help from above. And if you believe them then there is an indication. You can't have both weddings.
Personal feelings that prayers are answered are meaningless to me. Just like the faith that is revealed in excavations (there are no atheists in trenches). Evidence is only there when it is done in a systematic and controlled manner on large groups. Personal examples are meaningless. Unless these believers claim to have had a prophetic revelation.

It is certainly possible that any religious belief of any kind has added value over atheism, especially if it is a belief that is explicitly directed toward God, even if in our eyes it is wrong. God does not necessarily share the halachic definitions of idolatry as presented in the poskim.
There is no way to unequivocally prove the effect of prayers. You know the experiments and they do not convince you, even though quite a few of them point in that direction. You also know the statistics and they do not convince you, even though a large part of the findings there do not have a simple natural explanation (for example, a smaller percentage of birth defects in religious people). I do not think there is any other way to show this. Deciding that the default is not to believe in it until proven otherwise is an arbitrary decision, and to the same extent one can decide to believe in the Bible, the Sages, and tradition on this issue. If you choose not to believe, it is a personal position, but you do not present any argument that is supposed to convince those who do believe. Simply declaring "I do not believe" is not interesting.
It's true that everyone can hold onto different interpretations. That's also what you do when you prefer a natural explanation for statistical differences (even though there's no evidence that this is the reason) instead of a spiritual explanation. But you leave no real opening to be convinced of the opposite possibility, because you can interpret any data you're given differently.
If personal feelings are meaningless, so is your personal feeling that prayers don't help.

Indeed, it is true. What I am claiming is that the accepted belief has no basis, and therefore I have no reason to accept it. As far as I am concerned, if there is no evidence that God is involved, then He is probably not involved. This is my scientific view, and I do assume that it is correct. You are right that it will be very difficult to convince me otherwise. This accusation can of course also be directed at you. And yet science works quite well, meaning that the assumption that the world operates according to fixed laws works and proves itself every day. I do not know of many experiments that failed because of God's involvement in changing the laws of nature, but of course there may have been some that attributed the failure to something else. In any case, this certainly cannot be said about the assumption of God's involvement in the world (which works and proves itself).
The argument I present is the scientific argument. If science is right that the world operates according to fixed laws, then there is no involvement of God (except in exceptional cases perhaps, which is almost impossible to prove), and therefore this is the starting point I hold. And whoever deduces from it has the evidence. If this is not an argument in your opinion, then I have indeed not presented any argument. And neither have you.