He may've done so more than you may think actually. Immortality isn't necessarily good health. The implication is that while Mouku may recover flawlessly from instantaneous death, she must be subject to gradual changes, (otherwise how would she be able to grow properly 'n stuff?) Granted, if she ever dies, she'll simply just recover but perhaps only to the point where she's in a state of health like the one she was in before. This would leave her in a wretched state, at best being an old coughing hag and at worst, in a cycle of near simultaneous death and resurrection. Granted, the Hori Elixer would seem rather worthless with this interpretation and it doesn't entirely explain why Kaguya and Mouku both appear to be so youthful but it's an interesting thought nevertheless. At any rate moving onto the main topic of my post...
While I understand the whole calm, wise levelheaded History Teacher Keine meme has its basis in both her powers and a few teasing comments by Team Scarlet, it really has no basis in the actions of the well meaning but spastically presumptuous, rashly overreactive character and we actually see for ourselves in Imperishable Night.
Judging from what I've witnessed, I'd say she's vaguely aware of the generalities of what's going on but acts upon gut instinct without knowing any of the pertinent details. As a primary example, She's willing to take everyone and everything head on quite brazenly and without second thought if she thinks it might be a problem. In all of the I.N. story modes, she's the first to jump out and attack any random passer by with accusations of wrongdoing before they even get a chance to speak or go about their personal business. This includes even Reimu and Marisa quite ironically enough, two characters notoriously infamous for this sort of behavior.
She's an absolutely terrible judge of character as well, as it turns out that the people there, whichever team you may pick, are actually turn out to be there to help out rather than cause any harm of any sort. Further exemplifying her bad sense of judgement, she also rushes headstrong into these needless battles and they turn out to be more far more than she can manage to chew. She even goes so far as to assume Marisa's not trustworthy (okay, that might be a good call) because she's a Youkai (or maybe not…)
Granted she has good intentions in that she simply wants to come to the aid of defend what and whom she cares about, much like a superheroine. However her plans aren't always the most well thought through if you really take the time to think about them. Writing an entire village out of history to defend it from the night's terrors makes little sense considering you'll eventually have to write it back in for continuity purposes or face destroying the village forever. While it's possible that this could be simply credited as stalling for time, it doesn't explain why she's so protective of Mouku in the EX stage, as Mouku scarcely needs protecting with her powers of infinite resurrection.
Also before anybody simply goes ahead and writes me off, I'm not saying I want to see more "CAVE!" as this one dimensional interpretation goes a bit too far into the other direction in taking her readiness to attack to the exclusion of all other personality traits. The way I see it she's either really more like Gensokyo's equivalent of either Super Grover or Knuckles the Echidna than anything else and either of those could both be very entertaining roles to fill pendent on the nature of the work.
This goes especially so since canonically speaking, there's no shortage of intellectuals that could serve History Teacher Keine's roles while staying more in character so to speak. Take for example Alice Margatroid, Patchouli Knowledge or especially Hidea no Akyu, the child who spends her days peaceably sipping tea with guests and recording Gensokyo's actual history for the purposes of perservation. (Rather than distorting it to suit her whimsy like Keine, for whom teaching history would be both pointless and counterproductive to any good that could come of changing it how she does.) A rivalry between Keine and Akyu might be interesting too, come to think of it though, due to this conflict in the nature of Akyu's and Keine's wills and powers.
I'd go on analyzing others but most of them have already been delved into to some degree and I don't think I have quite the level of canonical knowledge of the games' actual events to do too much better than what was gone over once before. However if visual portrayals count nobody seems to draw Chen with the the sharp red fingernails like she has in Perfect Cherry Blossom. She's usually declawed by the artists to make her seem more innocent, harmless and cute.
