The fact that Conan barely has any flaws makes him an extremely unrealistic person. That's pretty much the main thing I have against him.
The only other pet peeve I have is the fact that he can also be careless in terms of his identity. There were countless amounts of times where he acted way out of line since he was assuming the role of an adult. I understand that it's hard for him to act as a child since he has an unbelievably high IQ and is much more mature than most people think he is, but it's almost as if he doesn't even care anymore, which has put him in unnecessary tight situations that he could have easily avoided.
However.....
As I've progressed in the series, I've also noticed that people have been much more lax about his actions (especially in the movies). I'm still dumbfounded by the fact that police members listen and even act on his instructions while not even questioning that he's still in primary school. But at the same time, this is also the reason why this show makes me laugh sometimes. It's hilarious seeing a first grader outwitting these adults. It's practically gold.