I was interested in the 'vanity' angle, as surely if you're saying that they don't to be filmed because 'they don't look their best' then that's like saying that some nights they don't make the effort to look their best for the fans. Which is surely short-changing the fans who've paid big bucks to see them? Why should I see an Eagle in (for example) an un-ironed shirt just because I see the show on Wednesday and not Thursday? I would have thought their insistence on sound perfection would apply to perfection in other areas of the show too.
I'm trying to remember if I actually read it, or imagined that, the Eagles filmed every show that they did. If they did, then the vanity issue wouldn't apply as a reason for them to not want the audience to film them. I know Barry Manilow films all his shows, and he's now made the archive public, though you have to subscribe.
http://www.manilowarchives.com/
(Not sure if you get to see the whole show, or just some of it.)
The possible drawback to the Eagles doing this sort of thing is that I think they have a more limited number of songs to choose from than Barry.
ETA: After doing a bit more
I think the 'videoed every show' came from something I read about them issuing the 40th anniversary DVD.