Like Steve said, the ATC does not appear to have owned the lithographs or the player rights. The contracts seem to be directly with the lithographers (Hylands makes no mention of tobacco at all), who are not all Knapp's American Lithographic, at least not directly (presumably they are shadow subsidiaries, designed to avoid anti trust law). The lithographic companies seem to be running product design and marketing for the ATC, they aren't just contracted printers.
Presumably there must have been an exclusivity contract for some period of time, otherwise we would probably have cards with all kinds of backs cashing in on the fad. This is deductive as no contract has surfaced. If the Cobb/Cobb card is from 1910, it would be the only copy of a T card made for the ATC sets made for a different or semi-independent firm between 1909-1912. At least, I cannot think of another example. Can anyone else? I would suspect they were tied to the ATC before public merger, like the lithographers, as more likely.
This postcard does not seem to pre-date 1915 in its origin. It looks like a much later postcard styling. Here is a copy with a 1951 mailing date:
https://www.ebay.com/itm/14479874516...a369%7Ciid%3A1. The others on eBay don't have mailing dates on the back. This postcard is in not evidence the ATC was manufacturing Ty Cobb brand in 1910. It is evidence that they owned factory 33 decades later, which we already know.