I live in Amsterdam and regularly go to the Leidseplein just because it’s fun to be there!
Outdoor cafes, street musicians, jugglers and acrobats, various cinemas around the square plus a major theatre, tourists to watch… ;-)
Even when most of the tourists have gone there’s always something to see and enjoy.
Tourist trap? No – only if you want to be trapped, maybe. The Leidseplein is for Amsterdammers, too – very much so. But it’s really very easy to avoid if you don’t want to go there (a map might help).
Nieuwmarkt? Not a “better experience” at all as far as I’m concerned – I only go there if I have to go to one of the shops. Not too much “Amsterdam life” there. Going to de Pijp is a better idea, or any other residential neighborhood really: “Amsterdam life” is to be found where people live.