Why the Chantry?
It’s pretty simple, actually.
The Templars are servants of the Chantry. They are given their duties by the Chantry, they can be commanded by the Chantry.
Yes, the Chantry seems very innocent and whatnot. We think it’s just another place of worship.
It’s not. It’s THE Church of Thedas. It’s an institution. It’s the silent power behind the sword and shield of the Templars.
You see it in the game. Elthina is the only one to rein Meredith in whenever the woman is making a fuss. Elthina is the only one who can get through to Meredith, even when the latter is showing signs of lyrium possession. Elthina refuses to involve herself verbally because she knows that her silence is all that is needed. A “neutral” stance in a time of oppression is a sign of agreeing with the oppressor. If Elthina verbally throws her lot in with the Templars, the mages will realize what’s going on. They will realize that they are being subjugated by the will of the Chantry.
With Elthina’s silence, they can imagine that she is ignorant of what is going on. They can imagine she is an innocent, when the very institution she represents spews lines about magic being an affront to the will of the Maker.
Everyone can pretend they don’t see what’s going on.
By attacking the Chantry, Anders attacks the value system. It’s very indicative of the way the game forces the player to make the final choice between the Templars and the Circle. There can be no middle ground. Elthina couldn’t play the neutral card forever, and with her dead, neither can you.
On a practical level, I can only imagine how tight the Knight-Commander’s security is, especially in the Gallows. The Chantry just proved a very easy target.
(For discussion on Elthina’s character, I really recommend Flutiebear’s posts on the subject. They’re incredibly interesting.)