Some kind of verification would be nice but no such verification is practical as far as I know. There are many technical and political reasons for this but the simplest would be that the most embarrassing candid videos would never be verified by the subject of embarrassment.
The only possibility is to cryptographically sign all videos using a White House private key.
Using a digest or MAC is not sufficient: it has to be asymmetric (i.e. public/private key).
Biden already comes across as a deepfake, or a tear-jerker actor in a commercial for senior insurance. I'm sure everyone sharing videos of Biden or Trump will check the SHA hash first.