Well that is what copyright upholds. The problem is there is no simple way you can get a license for the video clip from the rights holders. And then no sensible way of verifying that license with the platform.
While YouTube certainly is a public platform I think an interesting case could be made for videos with a limited popularity, say only viewed by family and friends.
There are a lot of opinions on how copyright should work, however little understanding of how it actually works.
There are possible tech solutions that could work with the current copyright system. The truth is both big tech and the rights collection societies like the status quo as they both profit off the actual creative works done by others. The actual battle is between who can get the biggest slice.