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.

Have you been to NY? It’s both. There are wealthy folks in the city but also some of suburbs are also some of the wealthiest places on the planet. Folks forget that you drive 30 minutes from the city center and you’re basically driving through neighborhoods of $1M+ homes that go on for miles and miles. It flies below the radar, which is precisely why so many wealthy folks hang out there.

in new york you're not remotely right.

the suburbs around new york are some of the richest in the world. Scardsale, every town near the ct border, rye, huge parts of li, montclair nj and the towns around it.

the average household net worth in westchester which is a huge county is $1m, thats on the same tier as wealthy parts of any major city.

Sames true of the suburban sprawl of the bay area and dc.

The suburban wealthy are a little more McMansion/nouveau riche.

Some of these people meet a certain definition of "rich", as in they never have to worry about money. Most suburbanites are not rich by that definition, there's a mix of negative net worth "keeping up with the joneses" types and the single digit millionaires who are a little less flashy and careful with their money.

A useful example - I knew a guy who lived in Naperville and owned an insurance company, drove a hot Jaguar and lived in a huge house. When the housing market crashed, he gutted it and sold off all the parts he could before the bank foreclosed on it.