Suburbs aren't a "not the cool, trendy area of a city" thing. They are a driving thing. They are completely different towns, not other neighborhoods in the same city. It doesn't count if you can easily walk the distance every day. Think more like 20km.
I mean, really, Millenium Park in Chicago is almost 2km long. The National Mall in Washington DC is about 3km. Central Park in New York City is over 4km long.
The US is, of course, different and drastically distorted thanks to building everything for cars instead of people. Human scale "cities" aren't 60km across. I'm not even sure they deserve to be called cities, but are rather a new sort of urban-ish agglomeration.
Before moving here I lived in Santa Monica. It was about the best I could do for finding a place where I could walk to anything interesting, and even then we're talking about fairly lengthy walks. When you get around by bike, walking, and public transit your sense of scale changes drastically, and you realize that most US "cities" built since 1920 are sprawling fields of asphalt with a building sprinkled here and there. That being said, I'm glad I was in Santa Monica where I was at least able to cycle to a reasonable number of things, and not, say, Northridge, or another place where the parking lots are almost as big as Dublin's downtown.
This is an American/Canadian vs Australian (and possibly European) thing.
From speaking to Australians, Downtown is the Central Business District, and everything else is a suburb.
Now that I've been here for a couple years I'd be fine with a nearby (within 2km of city center) suburb, but that's only because I'm familiar enough with the city that I don't need to explore it daily. If I had started off living in those burbs I would know the city much less than I do now.