Like a dash cam, they can save the footage only if there is a problem. Surely that would be much better than splicing together many third party camera recordings.
An NFL game has a ton of cameras, a ton of camera operators, those fancy cameras on wire things, an onsite editing crew, and an audience.
To get good recordings, you'd need to invest a lot of time and money, and very few of the recordings would ever be watched. Doesn't seem worth the investment given third party recording seem to turn up quite often and video isn't terribly necessary for the investigation.
It doesn't seem to be such a terrible idea, though i do stand to be corrected.