An ambitious smart young man who has clear life goals and the drive to serve the federation and push the limits of human expansion and discovery? His character is to show that all generations young and old are willing to further humanity’s/the federation’s goals of a peaceful coexistence and cooperation for greater prosperity for ALL.
Some of Wesley’s haters often come across as honestly jealous that THEY couldn’t be born into such a time and place.
Edit: furthermore, a character like Wesley is SUPPOSED to be the way he is. It’s not the actor it is the character. Young viewers were meant to look at him as a mirror to themselves, a way to imagine yourself in such a universe, someone to identify with and better put your mind into the headspace of the story.
While Riker is cool and funny, you are not Riker.
While Picard is cool and professional, you are not Picard.
The context that’s he’s supposed to be an annoying child? You’re really not getting that people can dislike a character that is “well done”. Yes, he’s well done… as an annoying child who tries to act like he’s on the same level as Picard and Riker. He’s the equivalent of internet comments.
He had a tendency to be immediately dismissive, impatient, and often outright insulting to other cultures and non-Federation types. The kid needed some humility, and often showed his age and closed-mindedness when he shut out ideas that didn’t come from his superiors.
That’s called “being a teenager” and having a character which the others can offer corrective criticism that is also a message to the younger viewers by proxy
Personally, I didn’t really mind Wesley in the later seasons and a few of the Wesley-centric episodes were pretty good. It’s really the first season, and even more so the first few episodes of the first season where he was annoying as all hell. Oh look, boy wonder saves the ship, yet again. Unfortunately that kind of seems to have stuck with the character, despite the show toning down his antics significantly after that.
The later Wesley episodes are good because you see him as a bright young man trying to do right in the world rather than a precocious kid who solves problems the grown-ups can’t.
The writers didn’t really know how to write the Crusher-Picard relationship(s) in the early seasons.