The influence of religious lifestyle and prayer in statistics

Why wouldn't Christianity and Islam have an effect? 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 don't 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 wouldn't rush 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 lie, and in all shades it is an obscene error. So if God helps the believers and the wrongdoers, then He can also help atheists. You are undermining your argument.
I disagree with your starting point. In my opinion, neither the sages nor all our predecessors had a 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.
Whoever wants to can hold on to all sorts of things that can be interpreted in all sorts of ways. For myself, I see no 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 do not. If you don't believe them then it is correct to say that there is no indication of help from above. And if you do believe them then there is an indication. You can't dance at two 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). There is evidence only when it is done in a systematic and controlled method 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 G-d, even if in our opinion it is wrong. G-d 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, although quite a few of them point in this direction. You also know the statistics and they do not convince you, although 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 something until proven otherwise is an arbitrary decision, and to the same extent one can decide to believe the Bible, the sages, and tradition on this subject. 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 stating “I don't believe” is not interesting.
That everyone can hold onto different interpretations is true. That's also what you do when you prefer a natural explanation for statistical differences (even though there is 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 are given differently.
If personal feelings are meaningless, so is your personal feeling that prayers don't help.

Indeed 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, that is, 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 that I am presenting 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 that I hold. And the evidence is there for it. If this is not an argument in your opinion, then I have indeed presented no argument. And neither have you.