IANAL, but I suspect there's a difference between uploading a video and saying "I didn't mean to" vs just copying massive quantities of data. For example offering a backup service.

A backup service means you're acting as a dumb pipe/dumb disk. You're copying bytes without knowing the contents at the request of your user. This doesn't apply when you're seeking out data to copy on your own. In some situations you're still fine, like if you're requesting files from public web servers. But I'm not aware of any protection that would blanketly apply to intercepted connections.

If you're operating a dragnet then really you're close enough to a dumb pipe. If you're doing very targeted capture then you can filter stuff out anyway.

Being a dumb service works because you're acting on behalf of someone else. Someone has to order the data to be copied, and they are the ones responsible for ensuring there is sufficient licensing.

I suppose you could send them a DMCA notice, but then you would have to prove such data existed.