Why shouldn't Christianity and Islam have an effect? After all, these are religions that believe in one God.
As I wrote, this effect can be explained in all sorts of natural ways, or one can suspect the methodology of the studies. So I am not relying on them to *prove* the power of prayer. But I can use them to *refute* the opposite claim, namely that no statistical difference whatsoever is seen between religious and non-religious people. The heretic may not be convinced, but the believer has no reason at all to doubt his belief on this issue.
A personal feeling in day-to-day 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 one kind of foolish followers or another without checking them, just as those who assumed that no difference at all would be found between religious and secular people were mistaken.

Christianity, at least in some forms, is idol worship, and in all its forms it is a deplorable error. So if the Holy One, blessed be He, assists idol worshipers and those who are mistaken, then He can also assist atheists. You are pulling the rug out from under your own argument.
I disagree with your starting point. In my view, even the Sages and all our predecessors had no basis for assuming Divine influence on life. So from the outset I do not give much credit to this belief, and I do not think it is presumed true until proven otherwise. On the contrary: for me, the default presumption is that there is no such influence until proven otherwise.
Whoever wants can latch onto all sorts of things that can be interpreted in all sorts of ways. As for me, I see no indication whatsoever of such a difference or such an influence.
There is no difference between using studies to prove the belief and using them to reject the refutation. Either you believe them or you do not. If you do not believe them, then it is correct to say that there is no indication of assistance from above. And if you do believe them, then there is an indication. You cannot dance at two weddings.
Personal feelings that prayers are answered are, in my eyes, meaningless. Exactly like the belief that comes out in excavations – there are no atheists in foxholes. Evidence exists only when this is done with a systematic and controlled method on large groups. Personal examples have no significance whatsoever. Unless these believers claim they had a prophetic revelation.

It is certainly possible that any kind of religious belief has added value over atheism, especially if it is a belief that explicitly turns toward God, even if in our eyes it is mistaken. The Holy One, blessed be He, does not necessarily share the halakhic definitions of idol worship as brought by the halakhic decisors.
There is no way to prove unequivocally an 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 have no simple natural explanation (for example, a lower percentage of birth defects among religious people). I do not think there is any other way to show it. To decide that the default is not to believe it until proven otherwise is an arbitrary decision, and to the same extent one could decide to believe the Hebrew Bible (Tanakh), the Sages, and the tradition on this matter. If you choose not to believe, that is a personal position, but you are not presenting any argument that is supposed to persuade someone who does believe. Simply declaring "I don't believe" is not interesting.
It is true that anyone can latch onto different interpretations. That is also what you are doing when you prefer a natural explanation for the statistical differences (even though there is no proof that this is the reason) instead of a spiritual explanation. But you are not really leaving any opening to be convinced of the opposite possibility, because whatever data people bring you, you can always interpret it differently.
If personal feelings are meaningless, then so is your personal feeling that prayers do not help.

That is indeed true. What I am arguing is that the accepted belief has no basis whatsoever, and therefore I have no reason to accept it. As far as I am concerned, if there is no evidence that the Holy One, blessed be He, is involved, then apparently He is not involved. That is my scientific outlook, and I do indeed assume that it is correct. You are right that it would be very difficult to convince me of the opposite. That accusation can of course also be directed at you. Still, science works pretty 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 the involvement of the Holy One, blessed be He, who changed the laws of nature, though of course it is possible there were such cases and they attributed the failure to something else. In any case, one certainly cannot say that about the assumption of God's involvement in the world – that it works and proves itself.
The argument 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 the Holy One, blessed be He, except perhaps in exceptional cases, which are almost impossible to prove, and therefore that is my default presumption. And whoever wishes to depart from it bears the burden of proof. If in your eyes that is not an argument, then indeed I have not presented any argument. And neither have you.