Shirou's Endings, Perspective One: The Three Acts of Man
The three routes of Fate Stay/Night can almost be seen as a trip from the Ideal end to a bit closer to the Cynic's side of the Sliding Scale of Idealism Versus Cynicism. Many fans describe it as akin to the journey from child to man.
In Fate, Shirou remains a child in many ways, facing the future with his ideals unexamined and uncompromising. His love for Saber is a very Fairy Tale like thing; indeed, in later remakes they are given a distant, but very fairytale like ending.
In Unlimited Blade works, we see a world view many would see as young adult—the view of a teenager who wants both the ideal and reality. Even though he's been shown the faults in his hero of justice ideal, he decides to pursue it anyway—a struggle, to be sure. Here, Rin and her Cynicism serve to balance his idealism.
In Heaven's Feel, reality and Sakura drive Shirou toward a more adult viewpoint. Here we see him (as a certain tiger once suggested) holding fast to what he can, and compromising when he must. This is more mature than he can bring himself to be in the other ends. His father was right, you can't save everyone—and so he will save what is important to him.
Shirou's Endings, Perspective Two: The Problem With Selflessness.
Many, both here on TV Tropes and elsewhere, have noted the fact that Shirou doesn't place much value on his own life. In fact, it is this that allows him his amazing use of projection magic.
But the three routes also show a change in this. In Fate, Shirou's ideal remains, as does his selflessness. He has the naive devotion of a fairytale hero, and it earns him (at best) a distant, fairytale, dreamlike ending with Arturia.
In Unlimited Blade Works, we see a Shirou who is less naive, but still clings to his ideal. In other words, what he experiences grants him enough self-awareness to acknowledge reality—and be stubborn in the face of it. Indeed, it is likely that here, Rin will be the "adult" in the relationship.
Finally, Heaven's Feel. While there is no doubt that Shirou is Sakura's hero here, we also see him asserting something he wants. In the better endings to Feel, Shirou puts Sakura's well being before his dream, before anything else. This is not merely limiting his dream to just being Sakura's hero. This is deciding he wants her. A selfish desire, even if his love drives him to great heroism—but that's actually the point.
While Shirou does noble things in the other routes, it is also there that most events that cause much of his labeling as a Canon Sue and/or The Scrappy come in. It's why Anime Shirou is one of the least liked versions—he's mainly based on Fate route, the one where he gets the least development. But by Heaven's Feel True End, we see a Shirou who can say, "No. I want Sakura to live." This does not doom him. It redeems him—asserting a selfish desire gives him a focus. Nor does it destroy his Projection ability—he can now direct his selflessness to something (and someone) he values, making him not a boy with a vague ideal but a man with a true cause—and a stunning skill to back it.
The Love Interests as Signposts in a Hero's Journey.