"But it's not logically correct to say that games = childish, then games = for children, even if the original premise (games=childish) is correct"
Perhaps not, but it does come across that way when people state it like that, be they right or wrong. :)
Perhaps you could expand what you mean by childish as regards activities, games included? Are you inferring that all gamers, by your definition, are child-like in their natures and lack maturity? I would argue that gaming, as a hobby activity like any other, has a range of people that participate, ranging from those who perhaps you see as less mature to those who you would see as mature? It's all perspective in the end, is it not?
Or is it a perception gained from certain sections of the media? Taken to obsession, as for any other hobby, this could be perceived as unhealthy by others and indeed has been, but this is not down to gaming itself, more down to the stereotype that exists for it, which is constantly reinforced by many of the press.
Alternatively are you stating a view more from a more social perspective in that if something that is enjoyed as a child it should not be carried through to when you become a "responsible adult"? If so, then I fear that I am a little behind in my social development by more than my hobby of gaming; I have yet to get sensible slippers, floral curtains and confess to not possessing a doily nor a special tea set (is that all I need to enter middle life in middle England or have I missed something? ;) ). I also do not own a knee length sensible skirt I'm afraid.
Note: To those who like sensible slippers, floral curtains and doilies/tea sets - I was just illustrating a point, and those things are perfectly lovely if you like them. Which I don't. :D