Uh, Archer being a symbol of what Shirou shouldn't become is quite a bit relevant to the fact that Archer wanted to kill Shirou. In quite a bit of ways, actually.
The thing is, Shirou imposes a black and white world view onto his mindset because he doesn't understand what ''saving'' or becoming a hero actually means and his only way of going about doing that is by dividing the world into black (bad, villains) and white (good, innocent everyday citizens) and then just ''eliminating'' the black side of the world in order to reach a fully white utopia where everybody is ''saved'' and happy. This is due to Shirou being ''violent'' and aggressive in his very nature and not being really capable of understanding the full range of human emotion - since he is a sociopath and has a limited range of emotions accordingly. Though, that isn't exactly Shirou's world-view in full extent. He has an understanding and more extensive grasp of the world, he merely imposes the black and white mindset onto himself because that's the only method he understands how to act upon.
The idea of Archer is what would become if Shirou started doubting his own world-view or rather the ''good side'' of it and becoming more and more focused on the black side of things, as well as becoming more cynical, maniacal and violent in everything he does after being shaped by such focus. Basically high-lighting Shirou's morally incorrect habits and doubt about his ideals. Archer imposes the black and white world-view so much that he completely forgets the whole reason why he does that until that actually becomes his automatic world-view. Archer wanting to kill Shirou is essentially a high-light of the fact that Shirou saves people for his own benefit and doesn't actually do it for others. It focuses on Shirou's ''truth'', i.e. selfish base behind his whole hero complex and actions.
He basically forgets why wanting to save everyone is morally correct and thinks of Shirou's ideals in a very machine-like and factual manner. If it's impossible to achieve, then you should chase after the impossible. So that's why he tries to kill Shirou. To try and create a time paradox, possibly erasing himself out of existence so he doesn't have to act as a hero anymore. Because it only brings him suffering and degrades his mental health further.
I'm surprised that gave her some character arc considering they said they won't.