That's completely unrelated to ancillary copyright (the Article 11 stuff), however. That's the DMCA. You still have the right to quote, but Youtube is not required to host your content for pretty much whatever reason they like, including just taking whatever causes the least trouble for them in case of a DMCA takedown notice.
Your right to quote exists vis-à-vis the copyright holder, not third parties. A frivolous DMCA notice may hypothetically get the copyright holder in hot water, but does not affect your contractual relationship with Youtube. And Youtube does not have to defend your rights for you, they just have to comply with the notification, takedown, and counter-notification requirements.