Honestly, I've never been able to get into FR myself. The Baldur's Gate games were fantastic, but I don't think the setting was the strength of those games - it was the writing and the gameplay. The setting was a decent backbone, but I'm reasonably sure they could have transplanted that story elsewhere and it would have been just as fantastic.
A few books caught my attention, but I think I only have ones till, the rest were handed off to friends. I dunno, nothing about FR ever really spoke to me. The sometimes rotating door of gods, the largely cookie cutter regions and factions, and overblown "Hero" characters like Drizzt and Elminster took away some of the mystery and majesticness of the setting - why is my band of heroes doing any of this when one of them could probably stumble across it and solve it without much issue? I mean, I know realistically that's not always possible, but as a player (and a GM) having those sorts of intensely, heavily touted heroic characters can really take away from things sometimes. You can say "Oh, well, they're off doing blah blah, they're not a factor", but in my experiance, people still think and worry about it - it ruins part of the illusion of being a hero, of being special.
If I had to pick a new "default"/flagship campaign setting? Greyhawk is probably the best one for the design goals of 5E - it's a loosely defined base that most every edition of players (except maybe people who started with 4E) are familiar with, with the ability to add or take away whatever you want while still having an interesting, solid base.
But that wouldn't be my personal choice.
As for Eberron: I think it's a really well done setting, but it's too far from the standard, pseudo-medieval milieu of D&D to be the flagship setting.
Eberron would be my choice. It may not be the most unifying location, but it is the most overtly dynamic setting, with plenty of various options that can be dropped in and out of a campaign as desired. And the fact that it's not strictly swords and sorcery, while still having more than it's share of just that, makes it something than can theoretically appeal to a large number of people, not to mention that it fits well with the idea of "familiar change" that seems to be prevalent in talk of 5E. It's also insanely easy to run any number of campaign styles in Eberron, and I think that would fit with the theme really well too.
Plus, damnit, Eberron was the most dynamic, interesting, flavourful setting since Dark Sun. I never liked the things 4E did with it, and I want it to live again in all it's glory.
EDIT: I will say that, unlike most of the rest of this thread and FR supporters in general, I actually liked what 4E did with FR. It was a radical shift, but it finally gave the setting something I felt was very unique and flavourful and fun to tinker with - it reminded me, in that sense, of Eberron's dragonmarks, or anything about Dark Sun. Granted, Dark Sun is a much, much more niche setting and could never be a true flagship product, but I think it has more going for it than most of the ones that could be.
So, why still support Greyhawk as a flagship setting? Because it's literally the opposite. It still has flavour and interesting bits and fleshed out, cool lore to it, but at the same time so much is so loosely defined and open that the possibilities are endless, more so than any other of D&D's "traditional" settings - and that's pretty damn cool, if you ask me.
That said, I also want a proper Planescape in 5E. Not this "Sigil is stuffed away in a book about the Planes" crap. Sigil deserves it's own freaking book. If Sharn could get one in 3E, I think Sigil could easily get one at some point in 5E.