It’s not so much about being built on a grid, but rather being built with a particularly high population density in mind - and further supported by a robust public transit network.

No, the problem is the network matters. When you can't get anywhere on transit you don't use it and in turn won't help improve it. I've many times looked at the transit options available to me and found I was unable to get my errand done on transit so I was forced to drive. One place I lived I checked and transit could do the job so I sold my car (but my wife still had hers because there were still many things we couldn't do on transit)