More rambling thoughts:
Dean’s dealing with feelings of monstrosity in 8.02—feelings that have continuously resurfaced for him at various points in various forms since season 2. Losses and traumas trigger these feelings for Dean. There’s the trauma-response to someone he loves dying which is “if loss keeps happening around me, I must be the cause of that loss” (either because he couldn't prevent the loss, the person died protecting him, or he's cursed). Then there's the trauma response from being forced to immerse himself in continuous violence to survive (aka, Hell and Purgatory). Hell and Purgatory both put Dean in situations where he has to be violent to survive, he adapts, he learns to find some form of comfort in that violence, and then he hates himself for it, even though the comfort he finds is false—just another trauma response born from the impossibility of coping without some sort of psychological relief.
In 8.02, Dean's coming off Purgatory (immersion in continuous violence) and loss (Cas) at the same time. Dean’s rough around the edges, still trying to shake off the battlefield mindset. Cas is pushed to the forefront of his mind as he agrees to help Kevin find Linda even though it’s dangerous and puts them all at risk (the same way searching for Cas put him and Benny at risk in Purgatory but Dean couldn’t let Cas go). In other words, Linda and Kevin remind Dean of himself and Cas (with Linda as the match to Cas) and that shapes how Dean responds to her.
Dean clearly likes Linda and thinks she's a badass by the close of their first conversation. He also jumps to defend Linda twice (once from a racist pawn broker, then again from Crowley's threats) but is restrained. In the auction house, something shifts after he's stopped by Samandriel, who asks him what happened to Cas. After learning Cas is dead, Samandriel says,
Too much heart was always Castiel's problem.
Something about this is triggering for Dean. After being reminded that Cas's heart got him in trouble, and that Cas is dead, Dean becomes colder about Linda.
After Linda gives up her soul for Kevin, Sam notes that the entire situation sucks, and Dean basically says, "This is the best outcome we could have hoped for I'm not complaining about it", which is very uncharacteristically unfeeling. Then Crowley possesses Linda, and Dean is ready to kill her in order to kill Crowley. Dean's struggle to pull back the aggression and "necessary evils" mentality that Purgatory brought out of him provides Crowley (and Dean) with evidence to support a false narrative Crowley sows to drive a wedge between Dean and Kevin:
I know we're not mates, Kevin, but one word of advice – run. Run far and run fast. 'Cause the Winchesters – well, they have a habit of using people up and watching them die bloody.
Dean immediately internalizes this, but there is no pre-existing pattern here—not in the way Crowley's implying. It's a gross perversion of the circumstances and feelings surrounding losses the Winchesters have been forced to bear. The Winchesters have lost a lot of people they care about—Mary, John, Ellen, Jo, Bobby, Cas... but describing any of those loved ones as people used and cast aside by Sam and Dean intentionally and unfeelingly is cruelly wrong, and Dean especially has struggled to weather these losses. Still, it digs into the "loss around me == my fault" trauma response stuck deep in Dean's bones (that has been recently rattled by losing Cas).
Dean admits after that he would have killed Linda for "the greater good" in order to kill Crowley.
DEAN: It was Crowley, Sam. No matter what meat suit he's in, I should have knifed him. I mean, yeah, it would have sucked, and I would have hated myself, but what's one more nightmare, right?
We see this "the ends justify the means" mentality early in the episode. At the Tran’s house during the initial Linda rescue, a demon starts to smoke out of Linda’s friend, and Sam does a reverse exorcism to put the demon back in her body because they can’t risk the demon reporting Kevin's location back to Crowley—it has to die. Dean kills the demon with Ruby’s knife, which also means Linda’s friend dies. Dean is the one who gets confronted about that, with Linda asking if he really had to kill her. Sam is obviously fully complicit, and Kevin supports their choice with an affirmative nod, but Dean is the one whose actions are questioned—who is asked to justify letting someone die to prevent further loss. He makes a similar calculation when Linda is possessed by Crowley—that further loss can be prevented through sacrifice.
I think aside from an atypical utilitarian mindset Dean's trapped in, another factor in Dean's choices with Linda is the parallel between Linda and Cas. Being ready to kill Linda is Dean questioning whether he ever should have looked for Cas at all. Fighting through Purgatory to find Cas was a foolish choice, made despite its foolishness because Dean loved Cas and would not and could not bear to leave Cas behind. But all Dean's determination—all the blood and sweat and muck—lead to failure, and then more pain and suffering. Too much heart was always Castiel's problem, and Dean wonders if maybe it was his own problem too.
