When you upload a video to YouTube, you agree that you own the copyright or are otherwise able to grant YouTube a license to do whatever they want with it [0]:
> If you choose to upload Content, you must not submit to the Service any Content that does not comply with this Agreement (including the YouTube Community Guidelines) or the law. For example, the Content you submit must not include third-party intellectual property (such as copyrighted material) unless you have permission from that party or are otherwise legally entitled to do so. [...]
> By providing Content to the Service, you grant to YouTube a worldwide, non-exclusive, royalty-free, sublicensable and transferable license to use that Content (including to reproduce, distribute, prepare derivative works, display and perform it) in connection with the Service and YouTube's (and its successors' and Affiliates') business, including for the purpose of promoting and redistributing part or all of the Service.
If you include others' work with anything stronger than CC0, that's not a license you can grant. So you'll always be in trouble in principle, regardless of whether or how YouTube decides to exercise that license. In practice, I wouldn't be surprised if the copyright owner could get away with a takedown if they wanted to.
Yes, this absolutely does not shield YouTube from liability from third parties, since the copyright holder of third-party content included in the video is not a party to the agreement. That's why they have a copyright notice and takedown procedure in the first place, and also the reason for numerous lawsuits against YouTube in the past, some of which they have lost.
To date, many Creative Commons licenses do in fact amount to "permission from that party", but if they start using DRM, those licenses would cease to grant YouTube permission.