I take a number of issues with this position not the least of which is that I don't think religion should or does hold total control of some of these ideas. Morality for instance, may not be completely understood using pure empiricism but philosophers, even some of them religious themselves, have been discussing moral claims without explicitly claiming a religious basis for them for thousands of years.
However, the philosophical problems I have with the NOMA isn't actually what I want to talk about here. I want to talk about why the argument and other similar arguments don't work on a practical level. The point of the argument is to attempt a diplomatic resolution with theists who have a problem with certain scientific discoveries. The idea is that rather than trying to convince them to abandon their religion (which isn't likely to happen) we can convince them that their religion and the science are compatible with each other. In a sense it's a noble goal but I would argue it is also almost certainly doomed to fail.
First we need to understand that an argument like NOMA is typically only used when addressing the fundamentalist type of religious believers. Of course it can be hard to define what a fundamentalist believes exactly at times because people don't always fit neatly into a box. There is a continuum between liberal and fundamentalist believers and even strong theological differences between some fundamentalist groups but there is one generalization we can make about fundamentalists. They believe that their holy book (the bible for Christians) is inerrant and contains true history, science, and theology. There is some debate on how to interpret context among Christian fundamentalists but they all generally agree that the bible is inerrant. This is important because more liberal believers are usually willing to interpret much of their holy books metaphorically so they usually have no problem with science to begin with, but fundamentalists have a problem with most metaphorical interpretations.
To understand why this creates such a problem for NOMA type arguments let's look at the concrete example of creationism. I choose this for several reasons, one it is one of the most common areas where religion and science conflict in the U.S. and two as a former fundamentalist I was once a young earth creationist years ago so I am familiar with both sides of the discussion.
To understand why evolution presents such a problem for fundamentalists we will actually start with Jesus and work backwards to the Genesis creation story. Most of my readers probably know that Christianity teaches that Jesus is our savior but if you haven't been steeped in Christian theology you may not know exactly how that salvation is provided, or for what reason. Jesus is suppose to save us from sin, but in Christianity the idea of sin is far more complex than just bad actions that you as an individual take. The concept is called original sin and the notion is that people don't just commit sins their very nature is corrupted by sin. Jesus' death is offered as a way to actually alter basic human nature and remove said sin nature.
Understanding that we can now look at Genesis. See the question becomes what gave us this nature, if it was built into us by god then his creation would not be perfect and also God would be blaming us for his failure. This will not do in fundamentalist theology so they have an explanation for this. Adam and Eve were created perfect but through an act of free will introduced this sin nature into human nature by eating of the tree of the knowledge of good and evil in the Genesis story.
My goal right now is not to analyze the rationality or evidence for such claims, but to elucidate as to why NOMA type arguments don't work on the one group of people that they exist to convince. The argument NOMA tries to make is that people can both believe in their religion and the science at the same time, but to a fundamentalist belief in evolution requires a denial of things they feel are intrinsic to their religious beliefs. It doesn't help that most fundamentalists view themselves as embroiled in a fight between the godly believers and the worldly unbelievers and they take a gateway drug approach to any ideas that they view as worldly. If a person drops even one of their core beliefs they take a step towards worldliness and who knows where that will stop. I am not guessing that this is what many fundamentalists think either, when I was a believer I read many books by theologians and preachers who made these exact arguments about NOMA.
The thing is in a sense one could say they are right, and they would use me as an example. To take the creation story as metaphorical one must deny the typical interpretation of biblical inerrancy and interestingly enough that was one of the first beliefs that I jettisoned on the way to becoming an atheist. It was over historical inaccuracies not evolution but as soon as I let go of the idea that the bible was perfect I began to accept other ideas because the evidence supported them and the more I read about these other ideas the less the bible made sense. My story is not unique either, few people just drop firmly held beliefs all at once but one piece at a time. This is why even though a non-overlapping magisteria approach may be more diplomatic than marching up to a fundamentalist and telling them science proves their religion wrong I'm not convinced it will be any more effective.