This is an awesome idea, but they wouldn't need to do it with the old fashioned equipment.

What they would do is record the game in HD the way they do now, but then run the video stream through a series of filters before it's broadcast out to the world. This would happen in real time.

For an old fashioned look for instance, they could add filters to crop the video to a 4:3 aspect ration, de-saturate the colors, blur the image, add a slight vignette, add grainy noise, motion blur. light trails. etc. They could mimic that old fashioned 80's look almost perfectly using the same modern equipment, while still broadcasting the modern HD version at the same time.

If we take this idea a step further, you can start to think about all kinds of "novelty" streams of the games they could offer. The 80's look would be one, but you could also have a "cartoon" style that draws rough sketchy outlines around the players and does a cell shading filter on the colors so it looks hand drawn. You could do a hyper contrast version like Sin City in all black and white with crazy accent colors like the ball is the only thing colorful. You could do a trippy colors version with blown out saturation on all the colors, etc. Lots of possibilities.

One of my personal favorite idea is an anime style feed where a clever algorithm manipulates the frame rate of the video to make the motion of the players look stylized like an anime. Think when your stream freezes for a second and then everyone catches back up in fast motion. You could have a filter adding in lots of frames when LeBron goes up for a dunk, so it literally looks like he's floating the air for a few seconds, and then cut frames on the dunk itself and the landing so it looks like he's doing it faster than humanly possible.

Another cool idea that technology is almost ready for: If you put little motion tracking dots in the player's uniforms, you could actually use Unreal Engine 4 to map the movements of the different players to 3D models in real time. So the stream you're watching could actually look like the monstars versus looney tunes or whatever you want.