But YouTube also has an unfortunate habit of deleting first, asking questions later. There was that Aussie teen a year ago who got hundreds of videos deleted, even from major networks' accounts, just by claiming copyright.

Not to defend YT, and I'm not familiar with that case, but that's pretty much what the law requires. If a DMCA notice appears on face value to be valid, it has to be treated as such. That's why there's a counter-notice provision, and criminal charges for making a false claim.

Now if YT didn't put the videos back up after a counter-notice or evidence that the notices were bogus, that's all their fault.

Some companies are better than others at enforcing these things universally; rarely, for instance, can you find any SNL online for more than a day.

Since you don't know the terms under which each video is created, much less the terms under which it was uploaded, there's really no burden on you. Nor is there any way of avoiding it, because you can't look at a video and think, "Hey, this just looks like something somebody did years ago but who is suddenly realizing it's worth money and may come after me".