That "私の力も妖怪による処が大きいです。" part, actually sounds more like, "Most of my power are those of the Youkai".
So yeah, like game2011 said, she doesn't sound so much as bragging, but simply saying that's the source of some of her powers. A bit different, but still matters.
How you take it is up to you though. You may also see it as her just being honest.
As for the next line, yes, Miko is right. She was tempted by evil power. (By the way, that "possessed" is kinda wrong too, "tempted" is more like it. "邪に魅入られた僧侶")
No need to question that further.
Regarding motivation though, I'm not really sure where it was said that this incident attracts those who wants power, maybe somewhere in the prologue?
Nevertheless, Byakuren can join in simply to calm down and solve the incident, like she did in HM. This is also true for some other characters.
Though I'd love to see a playable Shou, Nue, Murasa, Kogasa, Seiga&Yoshika combi, etc; I guess it's better to have the casts of MoF and prior titles, to join with the casts of HM. Therefore we'll get a sizable roster, but still well balanced. Though I want 1 more from SA and anyone from the fighters is fine too.
And another thing that also interests me, is character items.
That gap on the screenshot might be a Yukari item for all we know. It'd be interesting if we at least got more items to represent some of the characters that may not be included in the game. Though it'll be restricted only to some characters' due to the nature of their abilities.
Though one thing I kinda wish they add someday, is character support system or even straight out co-op. They've proven in HM that multiple characters on screen is possible after all. Just giving out ideas.