This situation suggests a lot of questions, but govloop (and YouTube) as organizations can do pretty much as they like, being as it's private property.
Clearly, there's the issue of copyright/ownership, and I've heard suggestions both ways, that government "stuff" is in the public domain, and that it's not. If it's the latter, then posting is copyright infringement.
But there's tons more here. If it is copyright infringement, who actually has the formal authority in government to have it removed? That's actually one of those thorny and ignored issues regarding ALL government participation online, because government staff enter in to contractual arrangements with third parties, without having the authority to do so.
Generally, I'm with Jim on the specific issue.
And, on an ethical level, which may or may not have legal force, and from an intellectual property creator, the fact that something appears somewhere in a public area does NOT automatically mean that others can do with it what they will. It seems that's the more common view these days, and it's counter to the spirit and purpose of copyright laws.
As an example (one I've used before), if the person who posted the videos posted them instead, on a raft of p*rn sites, or on extremist sites of various stripes, identifying it as coming from a specific agency, is that OK? Can you imagine the political outcry, not to mention public outrage at the "public money" being used to spread videos on terrorist or p*rn sites?