You're right, copyright laws were written ages ago when mass media was exclusively owned by big media companies. If there was ever a copyright dispute, the company's lawyers would meet up with the other company's lawyers and either settle or go to court, and both parties could easily pay for the legal fees because they were, you know, big media companies.
But nowadays everyone can simply upload something that can potentially reach billions of people, which is unprecedented in human (or legal) history, and the legal system simply hasn't caught up to this radical shift in the status quo. This is why Youtube has to compromise between the big media conglomerates with expensive lawyers and, well, the rest of us.