UC Berkeley doesn't have a right to distribute anything (they aren't "private people"). A private person has that right, but UC Berkeley is a public institution that doesn't have that right (despite the citizen's united ruling), and it is subject to ADA Title II restrictions. Private persons are only limited by ADA Title I (employment discrimination rules).
My understanding is that business/corporate entities do have some amount of 1st Amendment rights (e.g., speech and certainly press). IANAL; do state/local government institutions lack that right? If so, what about the professors working for them (who likely originated the content)?
It seems we have a case here of a law (the ADA) abridging a constitutional right. Again, we're not talking about a wheelchair ramp or a swimming pool or service dogs; we're talking about creative content. Courts seem to pretty strongly favor free speech over other interests even if those other interests are compelling (e.g., Citizens United).
The reality here is that the public is being harmed by the removal of and ceased production of this content. At a minimum, I wish UC Berkeley would ask the current DoJ to reconsider; they might come up with a different response than the Obama DoJ. (Plus UC Berkeley should cut off all interactions with Gallaudet if it can be found that institution was complicit in these actions.) But if there's a chance of winning in court on free speech grounds, UC Berkeley could do us all a service by getting that ruling. Of course, none of those actions coincide with Berkeley's own institutional political agenda, so they probably won't happen.