I think Shaw is interesting in some ways as a villain...here have another analysis.

Politics

Shaw is another example of 8th grade me slowly understanding how a lot of politics work. Especially nationalism. He's an isolationist who limits trade and a nationalist who believes in Trumpet superiority. I actually thinks this holds up well a year down the line. I mean, it is a pretty realistic antihero/antivillain motivation that has some parallels to real life. And you can sort of see where he is coming from.

Chip On His Shoulder

One of Shaw's main problems is his traitor dad. He wants to prove himself against him, prove he is more than a traitor. But this sort of comes more from a desire to prove he is better than his dad, than from true allegiance to the Trumpet Imperium as he eventually does betray them. He wants to show he is not a traitor. He basically has Spensa's motivation from Skyward, but turned up to 11 and given a bit of extra drama.

Elijah vs. Shaw

Elijah represents a lot of what Shaw is not. A perfect student with a perfect family. Shaw is envious of him and wants to take him down. This grudge starts at school and persists for Shaw's entire life. It is also what ultimately causes his downfall. He can not let go of old anger and move forwards for the good of the Imperium. His hatred of Elijah haunts him until finally he turns against his country and allies with an enemy just to take down Elijah. This is not entirely because of a pure grudge. There's also the fact that his nation turns against him for his actions and supports Elijah. He starts to think this is because of Elijah's plots. Alexander turns out to be a spy. Which further fuels his paranoia. When [x] happens it is the last straw, he figures why fight for something that doesn't believe in him when he can ally with an enemy and kill Elijah once and for all and maybe finally feel like he has a secure purpose where he is wanted?

How To Let Go

Shaw's main theme and a lot of the themes concerning the Imperium are that sometimes you just need to let go. Shaw refuses to let go of old grudges. He refuses to let go of war with the Saxophones and sovreignty over the French Horns. Elijah in contrast is able to let go of his hatred of Shaw and just go and kill him as a traitor, not as someone he hates. He gives the French Horns sovreignty and lets go of much of the Imperium's control over the land by freeing up trade and stuff. He starts to make the Imperium a better place to live in and a nicer place at the cost of their top spot of the pecking order. 

Death

Shaw dies at Elijah's hands probably, hated by most, a traitor just like his father. Which is sort of depressing now that you think about it, since this was the fate he really, really didn't want.

Hero or Villain?

What is Shaw's "classification?" Is he a villain for putting his country above all else? An anti-villain for having sympathetic (ish) reasons for doing so? He definitely turns more villainous when he turns to the Orchestra side, but what about before? Is he an anti-hero? Anti-villain? Morally gray? Sympathetic villain? Feedback is welcome.