I don't care how much of a genius Wesley is, or how many super-beings like the Traveller say he's destined for great things...no teenager belongs on the bridge of any ship, much less the flag ship.
Quite the contrary. I can see how a teenager, or even a child, could belong there. In the 24th century we are supposed to let go of our prejudices. If a kid turns out to be a genius, enters Starfleet Academy at the age of five, completes it in two years instead of four and shows unbelievable talents during his training, why shouldn't he be on the bridge of a starship?
Starfleet has been open to accept anyone who can prove that he is capable of serving there after evaluating them by forgetting any prejudices. We had a Klingon and a Ferengi serving on the bridge of an important ship, despite the fact that both races have really unwelcome qualities (for Starfleet, at least), probably even more than a child would have.
There was a 3 year old that was trying to serve as a medical assistant on Voyager (never got to watching the rest of the series). She could become a medical officer, or a bridge officer, she seemed to have the talents for both. No one was concerned about her age.
What about aliens that look like human kids even when they are older? Are they off-limits to the bridge, too? And by the way, Nog's height made him feel like a youngling in DS9.
I was angry when the crew of Enterprise had trouble carrying their duties when Picard was transformed in to a child. The choice of he child actor was bad for him to start ordering them around, but he had the knowledge, training, experience and qualities, yet he was forced to give up his position until he was restored. That was unprofessional.
The problem with Wesley is that he was
awful, annoying, and he wasn't believable at all. He was a nuisance. And
boring. A teenager or a child might have worked if his role was well-written and well-acted. Also, why did Wesley have to act like a stereotypical nerd, and why did he have to display complete social (and other) idiocy at some points, and ask the most stupid questions that one can think of? I'm a nerd, and even I'm not that stupid.
(Also, in the Abramaverse, James Kirk managed to become the captain of the flagship immediately after leaving the academy, and without having any experience, even though we was acting like a child just before going there. Now that's bad, but it didn't bother anyone.)