How in Heaven's name did it actually make the ByakMiko pairing stronger? (In spoilers, if you will.)
In HM, they are shown to be willing to acknowledge each other, letting the one who win to take care of the incident and both offered a solution to Kokoro. In Kokoro's story, they're even willing to fight together, alongside Reimu. The end of this incident also ended their conflict. In fact , it is Byakuren and Miko's solution that eventually helps lead to what can actually solve Kokoro's problem, having Kokoro not rely on the masks. The new mask from Miko helped make Kokoro realize that she can't have others make a mask for her(or made a mask from them) and Byakuren let her stay in the temple and taught her Buddhism to help her control her own emotions. They might not solve it completely though, since Kokoro still need more push to finally act. That's where Koishi and Mamizou kicks in.
Personally, I think the stars in HM is Byakuren, Miko, Koishi, Mamizou, and Kokoro. Reimu and the rest didn't do jack most of the time and were just there. Also, the only hint of Kokoro's whereabouts after the incident is either the Myouren Temple or the Hakurei Shrine. With all this said, fans just love to picture Kokoro as their child.
I admit, when I first heard Hopeless Masquerade would be about a conflict between Myouren Temple and the Divine Mausoleum, I was pissed as all Hell, not just because I 'ship Byak/Miko, but because I think, regardless of what they think of one another, such a quibble would be beneath them, and because religious conflicts in any form, in any medium, make my blood boil. I almost wanted to leave a pissed-off comment saying, "I want to ship ZUN to the Gaza Strip so he can see how 'fun' holy wars are!"
The thing is, while people call HM a "holy war" and a "conflict" and such, it's really just a bunch of people from different religious groups trying to impress folks by fighting; their goal isn't to destroy the other religions, its to gather faith, just like they've always done. I don't think calling it a "holy war" is as accurate as...I dunno, calling it a "faith-em-up" or something.