A few words about Vince Gill (aside from his touring with Henley et.al.)
A long long time ago, Gill was in a band called Pure Prairie League. It was best known, I guess, for the single "Amy." PPL was hardly a straight-forward country band; they were very much influenced by The Flying Burritos and similar groups. Both Henley and Frey were fond of them back in the Old Days (cue the victrolas and kerosene lanterns).
And Vince Gill has pretty serious cred: Eric Clapton has performed with Gill on several significant occasions (check out his take on "Tulsa Time" with Clapton); recording-wise, Gill has been backed by some pretty heavy folks, like Leon Russell, Waddy Wachtel, and James Burton. And - oh yeah - some guy named Leadon.
Does he belong in this current lineup? Well, he's not the first person I'd pick, but there are likely reasons that we, The Great Unwashed Public, may never know. The music business is like that.
Who would I pick? Well, back around 2005 or so, there was some talk of adding another vocalist/guitar player: a person named Sheryl Crow.
What?!? A female in The Eagles? I've heard Sheryl perform Eagles songs, and she just KILLS. I used to work for her on the road, heard her perform about a thousand times, and I feel that she could probably sing anything by anybody and make it sound really good. Henley knows her pretty well (not like that! Get yer mind out of the gutter), and I've seen the two of them sing duets. Next time you see her, ask her to do "Desperado." It'll melt the wax right out of your ears...