Devil's Advocate here. The phrase "owning the rights" applies to the physical world as well as to media. You "own the rights" to your land. You then get to restrict what other people do on your land. This is just as arbitrary and artificial a construct as copyright.
This is very true, and its shocking sometimes to realize how your "ownership" of physical property can be restricted by third parties.
The key difference between physical property and intellectual property is the ease of rights enforcement. It's easy to detect when someone has infringed your rights on your own physical property. The progress of the internet, on the other hand, makes it constantly harder to detect and punish infringement of intellectual property rights. It is still unknown whether reversing this trend will retard the potential of the Internet, but certain recent legislative attempts lead me to believe so.
I think that the debate will ultimately turn on the question of ease of enforcement. All the philosophical trappings of whether certain rights should be granted is irrelevant if they can't be granted.
You then get to restrict what other people do on your land.
Even this is far from universal. In many countries (Norway and Sweden for example), there are strict rules on what you can and cannot restrict people from doing on your land. You cannot stop me from walking across your land, camping on your land or foraging for food on your land (assuming of course I do these things in a reasonable and sensible manner, ie do my best to remain out of sight, don't camp too close to buildings, don't cut down trees or damage crops, don't stay in one place for too long etc, etc. ). Basically my right to free access to nature supersedes your right to do what you want with your land. The same sort of conceptual argument could be made about the "rights" to media.