I think that's definitely part of it... (the proving the smart kid wrong thing) but I also think you're right in pointing out the part where the show presents itself as being all technical and intellectual.
But I also think a big chunk of it is that it's more relatable than most other speculative SciFi. These characters are (mostly) humans, in the not terribly distant future. We expect things to be more or less a reflection of our own day and age, and so things that don't mesh well tend to get our attention.
I think another part is that in terms of sheer volume of content, Star Trek beats everything else. Hundreds of TV episodes, a dozen movies, a truckload of games... We know the Star Trek universe better than any other setting, so that it really feels almost real in a way that none other can, and I think that tends to make us a little less forgiving of anomalies and inconsistencies.
"He who takes offense when no offense is intended is a fool, and he who takes offense when offense is intended is a greater fool."
"Don't take refuge in the false security of consensus."