It varies by location and by what we mean by rich. In New York, for example, you're totally right. But for most of America the model is country club + suburb, 6,000 sqft house with a pool, big public school district that is very well funded, SUVs, &c. for the "rich".

And in some cities you actually have both. Where I live we have these big, wealthy suburbs (New Albany for example), Delaware County in central Ohio is one of the top countries by income in the whole country - all suburban. Yet we also have some absolutely fantastic and premier neighborhoods in the Columbus area with prices to reasonably match given the scarcity of actual neighborhoods and such, though I actually think the homes in these areas are a bit under-priced and the large suburban homes a bit over-priced.