This is how she phrased her demand. I don't think it changes based on our actions, but I haven't looked at the code so I could be wrong.
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I see things slightly differently in that I don't necessarily think the Templars are turning a blind eye. It really depends on who's doing the killing and why. If the combined governments of Earth are hellbent on enforcing the secrecy, it's entirely possible the Templars couldn't protect people from them anymore than they could rescue the Grand Master from the elves: they might make a good showing for a moment, but they'd be swiftly wiped out and then whoever they were protecting would be killed anyway.
Likewise, just because they aren't protecting humanity's right to know about Neverwhere, it doesn't follow that they aren't protecting humanity at all. It could be that the Templars view their role as protecting humanity against specifically supernatural threats. Things like demons or elves they'll fight, but what humans do to each other is beyond their mandate. This would certainly fit in with Kaija's insistence she needed to fight the Hellspawn in chapter 4. If true, we could debate the morality of such a view, but they may still provide a valuable service even if it's not the one we'd prefer.
Again, we just don't know enough to judge, at least IMHO.
Thanks, I appreciate your views as well.