I think rituals are important for human beings. It seems like we need some kind of rituals to keep us happy and contented. I like the Zen way of dealing with rituals by doing them but not really believing in them. It seems like a rational solution to the problem.
What I think he means here is there isn't assumed to be some magic or some supernatural "thing" associated with the ritual which is anticipated to do something special that could not be done without the ritual.
While I tend to share the basic skepticism to the supernatural, I don't think that stance historically uniformly true in Zen Buddhism, and in chanting, something different really does happen that is different than had one not chanted, like when you do anything no matter how mundane, it "changes the course of history." Whether that's supernatural or not is irrelevant to the fact that the chanting has some efficacy...subject to the issues surrounding "spiritual materialism" I'll discuss shortly. Now in the Rinzai school, in the tradition from Hakuin, chanting in Buddhist rituals is substantially more lively than in the Soto tradition, so maybe Warner's chants and rituals are just boring (I doubt), or maybe they're just done differently. But when we Rinzai folk chant, we show up to chant, and in order to do that there must be a kernel of belief in mindfulness to do that, and in that sense, we "believe" in our rituals.
About the efficacy of the chanting, my teacher had once said that something to the effect that Heart Sutra → Emmei Jukku Kannon Gyo1 → 無. What he meant by this is that the presence, the technique, and the mindfulness one would do as one practiced chanting the Heart Sutra, would by its nature be an expanded version of chanting the "10 Phrase Life Prolonging Kannon Sutra" (延命十句觀音經) which would be an expanded version of working with 無. Now of course Warner's school deemphasizes koan study, so maybe that makes his chants and rituals boring. But I doubt it.
As the Pure Land link notes above (with spelling corrected):
Towards the end of his life, the great Zen Master Hakuin (1689-1769) took an increasing interest in life outside the monastery and in the lives and practices of his lay diciples, government ministers and the aristocracy.
In a letter dated 1754 to Lord Nabeshima he tells of the virtues required of a good leader and the merits attached to recitation of the Enmei Jukku Kannon Gyo (Ten Phrase Life Prolonging Kannon Sutra).
Along with the letter he sent a copy of the Sutra and told Lord Nabeshima of the miracles that were associated with its recitation, both in China and Japan.
Hakuin expresses the hope that his lordship will recite it two or three hundred times each day and encourage his retainers, the uneducated and illiterate, to do so as well. He goes on:
'The reason lies in the testing. Give this to those who are seriously ill or have met with disaster for their consolation.
If it is recited with sincerity, miracles will without fail be accomplished and the person who recites it will be free from disease and attain a long life.
This applies to anyone at all.'
Now how to reconcile with what Brad Warner says with what Hakuin says? Well, Hakuin lived at a different time, first of all, and so wasn't exactly scientifically oriented. However, Hakuin's writing style tends to be a bit hyperbolic to put it mildly. He's say a lot of things to encourage people, from what I've read. (Then again, so does Warner...) And, full disclosure: when one has been sitting for a while this sutra is very useful to "direct the chi (気)" to where it can alleviate pain, and I'm not making this up. Sure, sure there's non-supernatural reasons for this, but it's not a placebo effect, it's more of the effects of exercise/yoga on fixing what ails your body, structurally speaking.
Now the thing is, you can't "get" this to happen by wanting, and of course getting rid of a gaining idea - the idea that you can get something out of the ritual - is probably completely impossible. That's because wanting not to want to gain or wanting to get even happiness and contentment from rituals are still things to be wanted to be gained from them. But given nonduality, some degree of avoidance of excessive scrupulosity is an accurate position to take, I think. Just bow. Just chant. And don't chase after thoughts about when you miss the right word, or have to take a breath and that distorts the rhythm and all that. Just go back to the practice...
1. 延 命 十 句 觀 音 經 (Emmei Jukku Kannon Gyo)
KANZEON NAMU BUTSU
YO BUTSU U IN YO BUTSU U EN
BU PO SO EN JO RAKU GA JO
CHO NEN KANZEON BO NEN KANZEON
NEN NEN JU SHIN KI NEN NEN FU RI SHIN