Regarding killing Joan:
In the text, the characters point out again and again, often by listing all of the threats to the Earth which they must address, that Joan is such a threat that must be addressed. With good cause, as she is slowly destroying the Arch of time, and she allows wild magic into the hands of Foul's minions. She threatens Linden's Army as well, again and again sending her caesures to menace them, goaded as she is by a Raver. And Covenant himself considers her a personal impediment, commenting that she "has me on a string" as long as she is unhindered.
So a resolution to the Joan issue is certainly much more than an ending to Joan's misery. It is rescuing the Earth. It is rescuing the Army. And it is rescuing Covenant. These are all motives for killing Joan.
Perhaps if they had more time, another answer could have been found. But Joan's death was the only answer that they had.
Which brings me to Covenant's insistence of responsibility for handling Joan.
Even if we have nothing else to go on, just the fact that Covenant insists he must take on Joan is enough to suggest that this has something to do with their relationship. But here, when explaining himself, he says, I can't go on until I've faced Joan. There is something of closure here. And he says, something more important than saving the world. This closure is that important to him. And, I've already sacrificed my daughter. Which hints at wanting to end things with Joan better than he had with Elena's ghost.In [i]Against All Things Ending[/i] was wrote:"Hell and blood!" Uselessly Covenant brandished his maimed fists. "I heard the Ardent. I know what’s at stake. But I’ve already sacrificed my own daughter. I can’t go on until I’ve faced Joan. Sometimes we have to do things that are more important than saving the world. Sometimes we can’t save anything else until we’ve cleaned up our own lives."
And Covenant's relationship with Linden is at stake here as well. Covenant tells Linden, "Don't touch me." Don't be close to me. Because, he feared what he was becoming - or what he might have to become. Dealing with Joan will change him, but he's scared about how it will. And, Everything that he required of himself while life remained in his body depended on his ability to grip and hold. He knew presciently that wielding the krill would be his answer to Joan. He knew he would slay her. This is what he feared would change him. Change him beyond Linden's ability to love him.