If I were thinking through the legal issues, I'd start by asking whether walking is permitted on the roads in question. Are people who don't have cars and don't have the money to take public transportation, for example, allowed to walk on the edge of the roadway to get from place to place? Or is pedestrianism prohibited on those roads? Obviously walking is forbidden on certain kinds of limited access highways and the like, as are bicycles.
The reason you start with walking is because I'd be willing to bet that runners are permitted anywhere that walking is permitted. Because from the standpoint of ordinary jurisprudence--as opposed to the rules and regulations governing competition--there's no difference between walking and running.
Do the roads in question have sidewalks? Communities might certainly be able to insist that no foot traffic can take place on roads that have sidewalks.
The analogy between cycling and running is slightly trickier. Bicycles are wheeled vehicles. Runners are pedestrians. Apples and oranges.
I would certainly begin with case-law, and I hope you get good advice here and elsewhere on that.
I think that gated communities and the like can certainly forbid running on their private roads. But you're quite right: public roads are public roads. My response to this person would NOT be a knee-jerk antagonism, but a willingness to take a hard, objective look at the laws that are actually on the books.
No need to submit until the law actually prohibits your presence.
At that point, of course, you can become an outlaw, and use publicity to your advantage. Post YouTube videos of "prohibited and illegal activies." Show Granny jogging in her pink track suit and show a (pretend) cop giving her a beatdown.