You'll find a lot of credible scientists who are big Trek fans because of the "fun" science and the adventures it allows. However, find me three credible ones who say the biggest features of Trek science are possible or even a fraction of possible. Transporters as in Trek: never. Artificial gravity as in Trek: never. Warp speed as in Trek: never. Almost instantaneous communications over light years as in Trek: never. Time travel as in Trek: never. If they aren't concerned about it, I'm not going to be concerned. If the Enterprise is suddenly used like a submarine and starts traversing a planet's oceans, maybe that will be the breaking point for me. Oh. Wait -- .
Seriously, Trek is meant to be approachable, mainstream entertainment. It's pop music, not opera. Within that genre, whether one considers it to be like The Beatles or the Jonas Brothers is a matter of personal taste, but those are the parameters of comparison. Neither are Verdi, nor should they even be compared to him. As others have said, Trek is no more (or less) serious science fiction than "Star Wars", "Babylon 5", "Battlestar Galactica", or anything else of that type.
As far as Trek becoming Harry Potter goes, I think Q pretty much took it there whenever he appeared, and he's a very popular Trek character. We are never to understand how he does what he does with just a wink or a wave. He is a true wizard. Odo may have crossed the line, too.