Personally I would say people should not be able to make money off someone else's copyrighted content without permission, but someone putting a lot of work into making a good video and having someone else claim all the money over 10 seconds of audio doesn't seem right either.

Recursive attribution (e.g. a profitable video used 10 seconds of a remix of a song ultimately belonging to xyz) would be fair. It should be easy for YouTube engineers to engineer profit sharing.

Say hypothetically that the song writer gets paid 100% every time their song gets played 100% in any video, why does copyright care at all? It's right to safeguard the primary creator's intellectual property to the extent of allowing them to claim share in profits, but it's strange to use copyright to restrict use.

Mathematically it would be simple, but I don't think it's as simple as just splitting based on time. They'd probably argue the song or movie clip added more value e.g. if I used a famous song in my 10 second introduction clip, that's a major branding move, is that still only worth 1/60 of the revenue from my 10 minute video? Or would I have many fewer viewers with weaker branding week on week?

There is also the right of the creator to have their work not used in ways they don't approve of. This a commonly heard about when politicians play songs at rallies. It's not just about the dollars.

Unless the purpose and value of the video is solely distribution of the copyrighted music it should be protected fair use. If copyright holders don't like it they are free to not license their music to be played in public ever.