This NYT piece provides a good way of thinking about how to add more homes to an existing area - specifically you can build *a lot* without increasing neighborhood building heights over what already exists (not that building up should be taken off the table, but it's one of those excuses people use).
It's New York City specific, but it shows how even in New York there's plenty of underutilized land.
And, even in New York, most opposition to new housing units is just concern about parking/car congestion. I'm not going to wave away parking as a legitimate concern (as much as I would like to), but parking people tend to go about it the wrong way.