Generally speaking, copyright is owned by the person taking the video. Hence, if you're in the video, you're generally not the copyright owner (unless you hired the cameraman).
The DMCA safe harbors do apply to direct infringement, as well as secondary liability. See the Senate report that accompanied the DMCA, S. Rep. 105-190, page 40 ("The [safe harbor] limitations ? protect qualifying service providers from liability for all monetary relief for direct, vicarious and contributory infringement.")
Sorry for weighing in here with a correction (rather than more colorful commentary), but actually, new works do enter the public domain each year under U.S. law -- unpublished works whose authors died more than 70 years ago. This is explained more fully in this law review article, and Section 303 of the Copyright Act.
I think Comment #26 might have you on this one, Mike. Section 412(c) does let a copyright owner that registers within 3 months of first publication get statutory damages and attorneys fees back to the publication date. It is interesting that at least one of the defendants is alleged to have distributed the film **before** the initial publication date listed on the registration. I wonder how that happened?
State schools enjoy sovereign immunity against damages claims (including for copyright infringement). So unlike private schools, UCLA need not fear big statutory damages awards. Makes it easier for them to stay in the fight and get a ruling on the fair use issues.
A big part of the problem here is the confusing number of ways your YouTube video can be taken down: could be a terms of use violation, could be a Content ID match, could be a DMCA takedown. It's a maze for most users to figure out. That's why EFF wrote and posted the Guide to YouTube Removals. Spread that link around!
minor correction
The DMCA safe harbors do apply to direct infringement, as well as secondary liability. See the Senate report that accompanied the DMCA, S. Rep. 105-190, page 40 ("The [safe harbor] limitations ? protect qualifying service providers from liability for all monetary relief for direct, vicarious and contributory infringement.")