I think it's reasonable to designate certain very-high-speed roads as "car only" when there are reasonable alternative routes for bikes. If there's a destination you can only get to by interstate I'd say "allow bikes." I'm a biker and strongly pro bike-rights but I recognize that certain exceptions are reasonable for safety & expediency. :)

So hypothetically, you wouldn't object (on "rights" grounds) if there were a non-interstate that was labeled no-bikes when there were plenty of alternate routes to everywhere serviced by that thoroughfare. Sounds reasonable to me.