IANAL… There are a few exemptions from copyright, parody being the most obvious.
The case against Napster, Limewire, etc. was simple - while they claim they are there to allow all sorts of open sharing, in fact they were basically baltant trading houses for copyright material. 99% of their material was popular songs or similar copyright material posted for trading without any permissions. According to lawsuits which took them down - they did not seem to even try to eliminate cpyright material, and in one case a company’s email suggested that the reason they were there was )obviously) to allow trading of copyright material without permission. So they were “inducing infringement”.
With youtube, you have the opposite. They don’t allow downloads (although there are ways to do so). They discourage copyright infringement and will actively enforce it (by cancelling accounts, as others mention). They will take down material when notified by the copyright owner with a DMCA takedown notice. A significant amount of their material is dancing cats, granny’s birthday or “He bit me! Oww!” which if violating copyrights (whats song does that cat dance to?) is only incidental and not an attempt to redistrbute the same product as the owner.
In fact with one episode mentioned in the court case, they were hit with a takedown notice for 10,000 items in 1 day and removed them all in 24 hours. Almost like the studio lawyers thought “ha ha, we’ll overwhelm them with this list and can use the excuse it took them too long to remove all the videos when we get to court”.
They even removed the “Downfall” parodies even though parody is an explicitly protected category in the copyright act (while “fair use” is not, it’s case law).
So really, here we have the sort of service the DMCA Safe Harbor was meant for. They are nowhere near just a means to bypass copyright. Much of their content is original or contains significant original elements. They actively discourage offenders. They take down any material that the owner complains about. They even took down the “Downfall” videos even though they were obviously parodies. They are NOT Napster.
Significantly, a lot of real profesionally produced material is permitted or even posted by the owners on YouTube. It serves a legitimate purpose, for free generally. When Viacom sued them, they did not seem to have any suggestions for improvement of the service, just “shut it down” and “pay us money”. Since it is not a blatant rip-off machine, none of those happened.
So if it were nothing but a bunch of unathorized film and TV show clips, like Napster was basically 99% unlicensed music files, they might have been closed down.