First let's start out by clarifying the point that whether it is good for people to have a literal belief in religious claims is a completely different question from whether those claims are true. Brooks makes an argument for the former, and does not address the latter point. Now on to the specifics.
1. Dogmatic religion motivates people to perform "heroic acts of service".
This is very optimistic. The other side of the coin is that dogmatic religion often motivates people to perform diabolical acts of terrorism. This isn't just the case for religions, but for any ideology that is accepted incontrovertibly. Nationalism, for example, can inspire both the very best and the very worst of people. This is a neutral point. I think people should be careful of any ideology that is inflexible and strongly believed. Especially if the source of the ideology comes from a centralized organization. Choose your ideologies carefully.
2. Dogmatic religion provides humans with standards of conduct that they wouldn't otherwise be able to provide for themselves.
Brooks says, "No matter how special some individuals may think they are, they don't have the ability to understand the world on their own, establish rules of good conduct on their own, impose the highest standards of conduct on their own, or avoid the temptations of laziness on their own." It would be unfortunate if Brooks believes that humans need their worldview and ethics given to them in an authoritarian manner in order for them to stick. If this is not what Brooks is implying, then this point needs further clarification. Clearly people need their parents to teach them good morals and a community that reinforces them, but this doesn't necessitate an autocratic religion. Laziness can easily be a part of dogmatic religion, because wherever there is a comprehensive and strict set of rules, legalism runs rampant. People no longer need to decide for themselves what is okay, because it has been decided for them. Those areas where the rules are not as comprehensive get very little attention, and people end up ignoring ethical questions in less regulated domains. Moral priorities correlate with the rulebook rather than with societal impact.
3. Dogmatic religion contains accumulated wisdom over centuries.
Religion does not have as good a track record of "out with the bad, in with the good" as Brooks wants to imply. If it contains accumulated wisdom over centuries, it typically also drags along with it outmoded and false ideas that stick around for no other reason than that they've been codified into doctrine. Where is the built-in self-correcting system in a dogmatic religion? How do false ideas get corrected and replaced with new wisdom to be "accumulated" as Brooks puts it? There is no such system in place. There is only the natural death and replacement of leaders to drive the progress. Each generation is trained to stick to what they have been told by the previous generation. New wisdom can only be accepted if it fits into the old worldview.
4. Dogmatic religion provides concrete assertions upon which to construct a worldview using logic.
It is true that a very strict religious creed makes the world simple and easy to reason about. Religion provides a nice package of answers to a lot of otherwise murky questions, allowing for one to build a logical structure on top of these underlying assumptions that can dazzle the imagination for sure. There's nothing wrong with using models to understand things that are complex. The problem occurs when the model is over-applied or does not fit the data. Does the model get updated? What if a better model comes along? In a dogmatic religion, the model is inflexible. The data must be reinterpreted to fit the model since the model cannot be updated to explain the data. This inflexible approach is not necessary in order to use models as an aid for understanding the world.
5. Dogmatic religion anchors people to their principals, making them less vulnerable to fad ideologies.
It is interesting that Brooks would use the phrase "mindless conformity" as an example of what dogma helps to avoid. Put in other words, conforming to a strict religion helps you not conform to society at large. One must conform, it seems, to something, and it may as well be a strict religion because those people out there are out of control with their sex, drugs, Rock 'n Roll, and Justin Bieber. How about instead of outwardly conforming with anyone, we look inside and figure out for ourselves what to stand for? I can stick to my principles without a literal belief that they are etched into the fabric of the universe and backed by a wild foundational story. My principles are informed by my upbringing and my community, and they have been and continue to be refined by my experiences and observations. They are my principles, and though they resonate and share a core with all good people in this world, they are my own.
6. Dogmatic religion provides unparalleled insights into life's mysteries.
Out of all the claims Brooks makes, this is the one I understand the least. He says:
Rigorous theology delves into mysteries in ways that are beyond most of us. For example, in her essay, “Creed or Chaos,” Dorothy Sayers argues that Christianity’s advantage is that it gives value to evil and suffering. Christianity asserts that “perfection is attained through the active and positive effort to wrench real good out of a real evil.” This is a complicated thought most of us could not come up with (let alone unpack) outside of a rigorous theological tradition.
I confess I cannot see what special insight there is to the idea that good can come from evil. Often the point is made that affliction can provide experience, opportunity, growth, perspective, etc. but it escapes me how this idea is unreachable outside of religious tradition. If anything, religious tradition would add to that some language about holiness, purity, being refined in a fire, etc. but even those are simple metaphors. Maybe there is nuance here that I cannot grasp because of my ungodliness? Somebody explain to me how strict religious tradition adds a depth of meaning to art that cannot be had without it.
7. Dogmatic religion's rigorous codes of conduct build self-discipline in its followers.
The example Brooks gives is a Mormon choosing to abstain from coffee. I never found it especially difficult to abstain from the list of proscriptions as a Mormon, because I found those commandments to be the easiest to check off, but maybe that is just a testament to my incredible self-discipline. Out of all of Brooks points, however, I think agree with this one the most, but with some provisions. It really depends on how this self-control is encouraged. If conformity to the religion's standards of behavior is brought about through guilt and shame, then this can backfire badly. Pornography is an excellent example of this. Addictive behaviors are fueled by guilt and shame, not deterred by it. People in strict religious groups have a lot of social pressure, and this can lead to stress and depression. As long as people freely choose to adhere to the strict requirements set for them, they can benefit from a community that respects and encourages it. Projecting these same sets of rules onto people outside of the religion, however, can get really bad.
In closing, Brooks shared a quick observation about how he observed that a blunt, right-and-wrong approach seemed to work better on an AIDS-ridden village in Africa than what he described as "vague humanism". I chuckled as I pictured in my head a small, soft-spoken, bespectacled philosopher trying to reason with a wild-eyed native about what actions will lead to the most beneficial outcome, and then some big, bible-thumbing pastor shoving the philosopher aside and yelling at the native, "THOU SHALT NOT!" I can see why the latter approach would be more effective.
The Book of Mormon musical may have been overly optimistic about people's ability to take value from religious narrative without complete literal acceptance of the truth claims, or maybe it was just projecting a hope that the world could be more like that. We have all been children at some point in our lives, and in the beginning we need a lot of yes and no, right and wrong, do this and don't do that. If our parents raise us well, we are eventually able to decide for ourselves what we should and should not do, and hopefully we can do that without harming ourselves or others.