I don't disagree with your last point, but the distinction between advertising signage/ graphics and product branding is material to the kind of control the owner can have. A sponsor or advertiser buys the use of time and space that belongs to or is controlled by others, while product branding goes on their own stuff before they sell it.

If someone puts up a billboard, they have no control over its replication in photos, videos, etc. The baseball games we see on TV are copyright exclusively by MLB: Citibank, Coca Cola or whoever have no rights to their rebroadcast or redistribution - or to limit it - no matter how big their billboards in the outfield are.

That's why I kept going back to the manufacturer. In the absence of the team for which the design was created, they would have copyright for the thing as a whole. As for the sponsors whose graphics serve only as advertisement, their rights are subordinate - if they exist at all. Of course if Team A&S/Mokambo was still alive I'm sure it would be a different story.