[A]ny genuine debate format I am familiar with involves a competition of arguments, with individuals (or teams) pitted against each other to prove or disprove their “sides.” And if you read Douthat’s article, it seems to me that’s exactly what he was talking about — “debating” religions in public to see which one is “best.” I’m having nothing to do with that. If you want to have civil public discussions that’s fine, but that’s not “debating.”
Now OK, Barbara doesn't want to play "my religion's better than yours" in formal debate settings, but I think her reason is because she doesn't want competition. Richard Dawkins doesn't play "my atheism's better than your religion" with the likes of the Ray Comforts and others of his ilk because he doesn't want to legitimize them by appearing in the same place with them. That reason for not wanting to debate a religion I can accept.
In polite company we don't go there, and there are various social and corporate taboos against going there, but...
Is one religion better than another?
As Buddhists, - and I include myself in this, because historically it's been true for me - we have been loathe to broach such topics because, perhaps out of a concern that we would create disharmony.
But historically this has not always been the case. Although the Buddhist to Muslim transitions are well known, when European colonists came to Asia the Buddhists did not immediately jump to become Christians, to say the least.
For example, Suzuki Shosan (鈴木正三), one of the more notable monks in the Rinzai tradition during the Tokugawa era, was the author of 破切支丹, Ha Kirishitan, translated as "Crush Christianity." It is hard to find on the web (I think Google Books has some books about that time that quotes this book), but if you read sections of this polemic by Suzuki, you will find arguments that PZ Myers and Richard Dawkins would approve. But that's not really my point, other than to note that I agree with a non-neglibible part of the projects of Dawkins and Myers in this regard (but I have some quibbles with them as far as "religion" in its most general sense, and other issues). No, that's not my point. My point is that in Tokugawa Japan, there was serious damage that Christian colonists were inflicting on Japanese society, including but not limited to, according to the Buddhists, the incitement of Christian converts to destroy Buddhist temples by the clergy.
Harmony was already in absence, and in the true spirit of budou (武道), dispute was employed as one means to try to re-establish harmony. Unfortunately dispute in itself was insufficient, and the Tokugawa shoguns resorted to rather brutal methods to re-establish political order. (The Christian converts who happened to be wealthy and influential threatened the Tokugawa shogun's rule itself, and given the history of the colonists, it was not an unreasonable fear that these converts might be a "fifth column.")
And so Shinto, Buddhism, and their offshoots (such as Konko and Tenrikyo) were allowed to exist and flourish without being challenged by the eliminationist creed that had caused particular problems elsewhere in the world. And (compare Japan's experience to the Philippines) eventually, when Japan re-opened to the West, they re-opened as a much more religiously tolerant society than most of the world (with the nasty Taisho/WWII nationalism as more or less an aberration).
Nobody, especially myself, wants a "debate" to get out of hand where we start crucifying Christians, as they did in the Tokugawa era (with the irony that crucifixion was unknown as a method of execution in Japan before the Christian missionaries, uh, suggested it). But given that Christianity, as well as Islam, are eliminationist creeds - their official ideology is that there is only "one true religion," you can avoid this point or you can take it head-on.
If you try avoidance, you cannot win, and it may come to the point (as it has in so many places) where eliminationist monotheists vandalize temples.
So you've got to strike a healthy balance. You've got to make sure you're not vandalizing any temples, yours or the Kirishitan's.
A koan by Seung Sahn goes:
A man came into the Zen Center smoking a cigarette, blowing smoke in the Buddha-statue's face and dropping ashes on its lap. The abbot came in, saw the man, and said, "Are you crazy? Why are you dropping ashes on the Buddha?"
The man answered, "Buddha is everything. Why not?"
The abbot couldn't answer and went away.
1. "Buddha is everything." What does that mean?
2. Why did the man drop ashes on the Buddha?
3. If you had been the abbot, how could you have fixed this man's mind?
Commentary: How do you meet the Buddha? Where do you throw away ashes? Its all very clear. Your correct function is always in front of you.
NOTE: There is an important factor in this case that has apparently never been explicitly included in its print versions [sic]. Zen Master Seung Sahn has always told his students that the man with the cigarette is also very strong and that he will hit you if he doesn't approve of your response to his actions.
Nonduality is not "everything is the same." Nor is it "everything is different."
And so it is with religious debates.
As to the question in the title, it's irrelevant, as far as the path that each individual must take.
Religions and other paths do have track records though, and they do have ways in which their myths, theologies, and ideologies can be used and misused, and some have been historically more involved in some nasty things than other religions.
I see no reason why that should not be fair game for discussion, especially given the existence of eliminationist creeds.