Shirou and Archer's ideal debate
The reason Shirou and Archer's "clash of ideal"' didn't really work is that Archer is shown to be a much more effective hero then Shirou is. The most obvious example of this is in all of the fights with Caster. Shirou, Rin, and Saber only manage to force Caster to run away once, while Shirou was nearly killed by Caster four times (at the temple, at the bridge, when trying to rescue Rin, and at the final fight). Shirou is saved in each of these fights by Archer, who never lost to Caster and is ultimately the one to defeat her. The only fights Shirou is able to win are against Archer himself, who is specifically stated to be holding back while trying to break his ideals, and Gilgamesh, who still needed to be finished off by Archer. So ultimately, Shirou accomplished nothing without Archer's help. It could be argued that Shirou's ideals weren't really about succeeding so much as they are about striving to save people, but if that's the case, why should anyone support a person who tries to help and fails over the guy who succeeds?
Shirou and Archer do not want to kill people. Their ideal is "save everyone" but Archer is super effective at killing, the exact opposite of what they both want. It's not a battle of physical might or results, as they both know that Archer is better, but which of the two viewpoints are valid. This has nothing to do with who's the better action hero, but why they want to be an action hero.
This would be a much more effective argument if it weren't contradicted by Archer's own statements. He flatly says that it might be necessary to kill the whole city to stop Caster, but it doesn't really matter. The point is that both are willing to kill to protect others. The only difference is that Archer has success while Shirou repeatedly fails, only winning (with help) after a time-loop loophole to gain Archer's skill without actually training.
From RAI,
To answer your questions...
Shirou's ideal is to save everyone. The problem is that he refuses to admit that the act is more important than the result to him. Kotomine trolls him in the beginning with this fact. That is, the "every Hero needs a Villain" comment. Or to put it another way, if you wish to save others then you must have a wish for others to be in danger in the first place.
Shirou is basically a machine for justice. Everything in his life relates to that desire. Ayako calls him out on it. Notice how he only reacts positively when he is in the act of helping others.
Archer is what Shirou becomes if he was never questioned on his dream and how unreal it is. He faced a hard rock of reality and broke himself upon it. Archer, unlike Shirou, prefers results over the method with the reasoning that if he can't accomplish his dream with his methods, he'll settle for the next best thing. Which is to say sacrificing the few for the many, which is what Archer hates, it's just that by the time he is summoned, he is so cynical and jaded that he has become apathetic to it.
This is a bad thing, even ignoring the fact that he does not view the world like a normal person. Kiritsugu faced this problem and was broken in the end too. Archer is unhappy with this settlement because he still wants everyone to be happy. This is why, like Kiritsugu before him, he banked everything on the idea of a miracle solving his problems, because he could not comprehend any way to realistically accomplishing this himself.
The overall message is that the means matter just as much as the end. As for why Archer doesn't just snipe or assassinate Shirou.
Until the confrontation at Ryŭdo Temple, Archer still isn't sure if he wants Shirou dead or if he can talk his way into what he wants. The final straw was Shirou going after Caster when he stood no chance, wasting the life he had saved. He then decided that if Shirou was going to die anyway he might as well do it himself.
He's also doesn't believe his time paradox will work, since Heroic Spirits, and by extension Counter Guardians, exist outside of reality. However at this point, he's ready to at least take his anger out on the source of his problems. Himself, Shirou Emiya. Of course, he has a tendency to lose his cool when dealing with personal problems. Even as Shirou he had that problem. It's noticeable in Extra where he shows up but there's no Shirou, so he acts completely different.
The other reason is that Archer actually has a plan for if he wins and Shirou and/or himself are dead. He plans for Rin contracting with Saber and winning the war after he's dead. The main problem was that he didn't expect her to contract before he accomplished his goal.
Hope this clarifies things, also this kind of argument is better off on the headscratchers page. Just so you know.
Rai out.
