Interesting inquiry, Danny. I'm going to bypass the whole question as to whether your premise about 4E being more successful as a non-D&D game because I don't really think that's the point of your original post. However, first I'm going to comment on this:
I probably wouldn't ever have looked into it if it hadn't been the next edition of D&D.
I'm not saying this because I believe it's a bad system - because I actually think it's a good system. I'm saying it because I am aware that there are dozens of equally good or better systems out there that I've not yet looked into simply because they're not as well known as the D&D brand.
Yes, I agree with this although it is more than just because of how well known the D&D brand is, it is being part of the D&D universe, the iconic themes and tropes of D&D. There are other fantasy RPGs with systems that I actually prefer to
any edition of D&D, at least on an artistic/design level (say, Talislanta and Ars Magica, for instance), and if I were to name my top five pre-published fantasy settings I'm not sure if any TSR/WotC worlds would be on the list (or at least not in the top two or three, which would be Talislanta, Earthdawn, and Shadow World), but the thing is that I grew up on D&D, I was imprinted with D&D (AD&D 1E, to be exact), and I am habituated to the tropes and themes of D&D. I've played and enjoyed other RPGs, but I always end up coming back to D&D because...it feels like home, I guess.
So 4E as a different fantasy RPG wouldn't have caught my eye as much; I might have bought the core rulebook for collection's sake, but I probably would have focused my gaming time on whatever the most recent edition of D&D was.
Back to Danny's inquiry. If I understand your question, you are asking what would we do with the 4E rule set if it wasn't died to the D&D legacy? If it was just another fantasy RPG? And, in light of what I wrote above, what sort of game would have caught my eye?
I might do is try to create a non-anime Exalted type game, a
Gonzo Fantasy RPG, so to speak. Like the idea of Exalted but dislike anime/manga? Try Gonzo Fantasy RPG!
As for specifics, I like a lot of what you've already said - do away with alignment, the class structure, maybe even skills and feats. Strip the game down a bit and build it back up.
There would be no classes, but instead something like "power paths" - I'd take the power source idea and exploit it. If it truly was an Exalted-esque Gonzo fantasy game, then all characters might be inherently "magical" - or at least imbued with a power source, which would be akin to different energy types or signatures with some kind of mythological theme. Maybe all characters have the blood of a different lineage of primordial races: The Blood of the Warmasters (Martial), Wizard-Kings (Arcane), the Dark Ones (Shadow), the Enlightened (Ki), etc.
All PCs might have a score in different power sources, and thus could "multi-class" by developing different power sources with experience points. Each point in a power source opened up new powers, sort of like levels in D&D.
Most skills would operate through ability scores, but PCs could specialize in specific skills, or create their own specialties within the purview of an ability score.
I might change ability scores slightly, maybe differentiate Wisdom into Willpower and Perception.
I'd kill Feats and replace them with Talents and maybe add Flaws or something similar to the Greek notion of Hamartia (your tragic flaw), where the player and GM have to craft a significant defect to the character...
And so on. I'm just making this up as I go along, so take all of it with a grain of salt...