That's not low-density. Maybe not everywhere is as bad as Manhattan, but that is indeed very dense. Note that you twice described your ideal solution as dense:

"a dense mix of single family homes, shops and apartments"

"a healthy mix of uses and density centered around transit."

Not a big fan of that kind of density, nor are many others. I'd at least like to have my own backyard, and not have my neighbors peering in from a second story.