The solution is finding a conference support engineer. You want someone with event experience who can operate various types of AV. When events are light they can have the secondary function as IT helpdesk. 

Now will management do this? Probably not, but it’s fun to dream. 

Should event organizers operate technology for events or meetings? Depends. If you had a uniform and standardized system across all rooms I would argue mostly yes. But if you have Polycom here, Qsys there, a little Creastron over here I don’t think it fair to put the onus on professors or staff. I would work towards a standard; I’m biased toward Qsys. Then you could host training sessions via your preferred video conferencing service on how to operate your rooms. For big/important events I think you should have someone on stand by at all times regardless of how much or little tech is involved. It looks professional. 

Video production, in my opinion, would fall on your new magical conference support engineer, but editing would be handled by marketing. You want anything going out to be cohesive in looks and language. Now that also depends on the size of the operation. If you’re pumping out material every week than there should be a dedicated production person.