I think that public spaces are a great thing, and I'd be happy to see them make a comeback. During my (brief) time in Europe, it was incredible to go to a store, pick a cafe and just sit outside with friends while there's a lot of other people doing the same thing. In this case, it's probably a difference in city planning that enables it to happen in Europe as opposed to America. That said, it's still possible to create these areas. I feel that they (should) promote community, support local businesses, and also promote activity. If you do it right you should have multiple public areas within walking distance of each others.
Unfortunately, I suspect that the biggest factor is planning around automotive traffic v. planning around pedestrian traffic. That sort of public space is harder to maintain when people have to find a pay for parking for it. That's unfortunate because it's going to be very difficult to undesign our reliance on cars.In this case, it's probably a difference in city planning that enables it to happen in Europe as opposed to America.
One big problem that has happened in my city is the type of crowd that has become attracted to these public spaces. My town has a very nice historic downtown that has a waterfront and many areas to walk and shop. Over the past 10 or 20 years though the area has seen a massive rise in the amount of bars, sleazy clubs, and tattoo parlors. Thanks to this near every time I go downtown it is mainly occupied by drunks and street crawlers making the area unsafe. Public spaces tend to breed conditions like this and limitations on what kind and the number of shops could help to promote positive, welcoming environments for people to visit. Imagine if 5 or 6 bars popped up in a mall, who would go there?