I'm suggesting not limiting foot and bike traffic just because we choose to limit car traffic. There are lots of routes between places in my town that would be much more direct, and safe, on a bike if there were small connecting paths between neighborhoods, including those built at different times by different developers, instead of being forced out onto the arterials.

Yeah, I think that's the part that I was suggesting should be "banned". All neighborhoods should connect with all adjacent neighborhoods via pedestrian or multiuse paths. And yes, that means across arterials as well -- either have an official surface crossing with appropriate traffic calming measures / pedestrian islands, or build a tunnel.