He does not act like a cop, you’re right, at least any of the ones I have known. Luther does get into workplace politics, though, and then exaggerates them, and that is going to resonate with a heck of a lot of people.
Alice works because she taps into a vigilante impulse (she routinely defends the good guy) while at the same time keeping some distance because she’s a sociopath. She’s brilliant, accomplished, posh, and very bad-ass.
The show meets people where they are.
You might have a backstabbing “friend” for example. Watching a worst case scenario with the “friend” killing one’s ex, framing one, and trying to get one shot dead by one’s co-workers... well. It’s hard to feel sorry for yourself about your own situation after running through a scenario like that :)
Alice doesn’t work on any level. Her character is preposterous and so overreacted she makes Anthony Hopkins playing Hannibal Lecter look restrained by comparison..