The thing is each of these settings had compromises made in order to facilitate the 4e mechanics (as an example Dark Sun wasn't as gritty and deadly).
Meh. Dark Sun had a terrible time in it's native 2e because of the lack of clerics, and thus healing. A non-issue in 4e, which had 'Leaders' of every Source to fill the void. Dark Sun simply worked better in 4e. From what I recall of the Dark Sun Encounters Season, it was plenty gritty and deadly. You weren't going around looking for treasure, you were constantly struggling to get enough food & water to survive - and the combats were brutal, even with the PCs slightly powered-up by feats.
Deadly just isn't an issue when you have solid encounter design, you can dial it up as high as you need to. Gritty is mostly a matter of DM presentation.
I also don't consider re-skinning anything like the type of customization we were speaking of in 3.x.
Re-skinning was a 3e thing, it just applied to less. The level of customization in 4e was pretty high, though, apart from that, just a good deal less abuseable.
In 4e can I play a balanced fighter whose equally adept at the sword and bow?
Sure. You just have to write 'Ranger' on your character sheet. The fighter class isn't stretched so thinly over the martial archetypes in 4e because of the addition of the Warlord and the non-casting status of the PH Ranger. Fighter doesn't cover as much because it doesn't have to.
4e did still suffer from D&D's long-standing inability to give martial archetypes their due, though, even if it was to a lesser extent.
(As an aside, though I think someone else hit on it; yes, there is a "sword and bow" fighter, and it's the Slayer. Most of their features work just as well with a bow as a sword, and I've seen that put to very good use. It was later in the game's lifespan, but it's out there.)
Yeah, I even played one. (it's almost off-label, too, the class is very clearly designed to be melee-oriented, it's power attack doesn't work with a bow, but it's DEX-to-damage striker feature makes it a natural DPR machine when using one) But, it's Essentials, and we were discussing the balanced version of 4e. ;P
I also have my doubts about your "magic rich" setting working since number and power and availability of magic items was tightly regulated in 4e.
There's a guideline, but it's really only important in the terms of enhancement bonuses. You can flip the Inherent bonuses switch, and have a low/no magic game below those guidelines, or you can use the guidelines and have a decidedly high-magic game. Since, at that point, you have magic items that can be readily made/bought or even dis-enchanted and transferred or transformed from one sort to another. That's about as magic-rich as I was thinking. You could heap even more magic on the campaign if you wanted, maybe by a factor of 5 or so before too-high-enhancement bonuses started cropping up (and, if you wanted to go there, could nix various feat taxes to bring the math back into line).
At what cost... if I don't like the AEDU structure can I pick a character that doesn't use it?
You might as well ask what if you don't like hps or don't like rolling a d20 or don't like classes or levels. It's just an underlying structure of the system. It really has very little bearing on the character you can play or what you can do with that system.
If every class has to use powers and they have to be in an AEDU structure... how is that not rigid?
It's not rigid because the resulting balance leaves the game very flexible. You can vary pacing radically without suffering from class imbalances. That's flexibility, not rigidity. You can focus a campaign on certain types of characters or themes, you can have a campaign with or without deities, with or without magic. You can focus on one pillar over the others. Players can play the general concepts they want, because the broader archetypes modeled by source can handle many roles, /and/ because no role is absolutely vital. With a robustly balanced game you can do more /without breaking it/.
Now, if you were on the other side of the publication model, you'd notice something like rigidity. When Mearls decided to let some cracks into the neatly-balanced AEDU system with Essentials he called it "opening up design space," because robust balance requires a lot of design discipline. It's just harder. If you've ever tried creating a new class for 4e, for instance, you probably found it /very/ difficult compared to creating a 3e class (especially a caster, where you could just build a list from existing spells).
But, I don't think you were talking about homebrewing classes.