I always assumed that the Phoenix
was a secret U.S. government project that Cochrane and Sloane were in charge of until something went horribly awry during WWIII. Heck, they have access to a former nuclear missile. In the aftermath of the war, they probably figured that they could finish it and sell it to the highest bidder.
The narrative that the 24th century had is just too perfect. There's just probably a lot in the background that we still don't know about.