Promotion of the useful arts and sciences is the language the framers used, but the framers didn't know much about economics. The economic function of IP is to address the market distortion that could result if people were permitted to free-ride on the capital-intensive R&D or other people.
The economic definition is also far more useful. Who knows what specific policies will result in "promoting the useful arts and sciences." Preventing free-riding is a more concrete issue that can be addressed with specific policies.
If you're making an economic argument to justify the existence and nature of copyright law in today's society, which in itself I have no problem with, then surely you also have to consider that exponential propagation via entirely non-commercial sharing could still severely undermine the commercial value of a work. If that sort of work then becomes prohibitively expensive to create, doesn't that mean copyright is not doing its job as an incentive and everyone loses out? After all, if I make the next great independent TV show, it doesn't much matter whether 99% of my potential market sees it on YouTube or just torrents it, if either way those potential customers no longer pay me in return for enjoying the work. This seems counter to your original position that the economic function is copyright is about rival producers and not consumers.
The economic definition is also far more useful. Who knows what specific policies will result in "promoting the useful arts and sciences." Preventing free-riding is a more concrete issue that can be addressed with specific policies.