Which government institution should decide on the video codec for everyone to use? How would they know when it's time to switch? I cannot see that working any better than MPEG and MPEG-LA.
That institution probably does not exist. The Federal and State governments are still trying to figure out what an Internet is. This isn't even on their radar.
For codecs, the government would not even need to regulate. As long as the offerings are good, the industry would be compelled to adopt them because of the lack of licensing costs and for the long term stability. Continuous R&D would still be necessary.
Well, for cryptographic hashes the relevant institution in the US is NIST and they switch at a point when there start to be worries about the previous hash being subject to successful attacks sometime in the future.
There's no reason in principle that the same approach, again with NIST as the relevant institurion, could not be used for video codecs.
It would be better than the MPEG-LA because the patent situation could be made much simpler (e.g. automatic patent licenses would be granted to all implementors of the standard).