tbh, I think that they're a completely separate issue. I know there are rules pertaining to games being blacked out, based on if the game is sold out, what media market you live in, etc. There are no rules that I know of about press conferences. I'm a fan of the Buffalo Bills, for instance, and they live-stream most if not every press conference, and have them uploaded pretty shortly thereafter on YouTube. There are blackouts for streaming games in the NFL, but they still have a media department that can do this.
I think the Spurs team media department just sucks/is embarrassingly understaffed and underfunded for a professional sports franchise. Local media was sharing quotes, pictures, and video clips from the press conference on Twitter, as well as the players shooting around a bit on the court in their new uniforms. It took a solid 6 to 7 hours after local media was doing this for the Spurs to even put out a tweet of the guys on the court/with their new uniforms/numbers, and there's still no clips or video of their presser to be found on any of the team's official platforms.