While I could see the entire escape from goblintown being done as a skill challenge, I don't see 4E doing any better of a job with LotR than any other edition.
The skill challenge doesn't allow 4e to do anything you can't do without them almost as well. What it allows is a new DM who understands them (and WotC's guidance is not good at this) to cope with unfamilliar situations and off the wall PC plans almost as easily as a veteran DM.
I've written a few paragraphs today that were missing from the skill challenge guidance and explain how it was meant to be used.
The reason 4e does better than any other edition at LotR is that the mundane/magical split isn't as harsh, and there is no dependence on a cleric. Most of the time in both the Hobbit and the Lord of the Rings, the company is without Gandalf or anything resembling a caster. Which means no healing. And half the game gone. At least in any editions before 4e. 4e you can keep the high action playstyle and not have a single caster in the party. It makes for an interesting dynamic, but not a gaping hole.
(There's a simple house rule that really makes LotR work better in 4e than it otherwise would - Extended Rests may only be taken at a place of safety - somewhere like Rivendell or Lorien).
D&D was inspired by a myriad of sources - LotR, Lankmar, Conan, Dying Earth, etc. D&D's strength isn't in how well it models LotR, but how well it lets you play any sort of fantasy.
Pre-4e was poor at most of those. Lankmar? Fafhrd and the Grey Mouser? A thief who was an apprentice magician and a match for his barbarian friend with a sword. And no mages or clerics with them at all. 4e: Thief with Ritual Caster. Dying Earth? A setting where great archmages could only memorise half a dozen spells at a time but were generally extremely
physically as well as magically capable? Despite D&D Wizards being called Vancian, that really
really doesn't fit. 4e: PCs with abilities above and beyond the norm and you pick ones with only a light dusting of spells? Sure.
I think 4e Martial classes can do very well emulating the feel of the LoTR movies. The books, not so much.
Having been playing a 4e Martial Only LotR campaign for a lot of last year, this is accurate

And it worked pretty well.
If I were doing it, I'd suggest martial classes only, but allow a few arcane and divine multiclass feats, eg Aragorn might be a TWF Ranger with Multiclass Paladin, Elrond a Fighter with multiclass Wizard. Ritual caster feat lets Elves et al use non-battle magic. Gandalf type wizards as PCs would be a problem, but there is no indication of mortal human Wizard types in the books. In the books Gandalf had an at-will burning hands type power, but that derived from him being an Angel wielding the Ring of Fire.
Pretty much what we did (with a broad-ish definition of Martial to include a Skald and a Hunter Ranger).
LotR has one issue that doesn't work very well with D&D though: narrative combat. It's different in the movies somewhat, but in the books, most "combats" were actually puzzles.
This is a very interesting point

And one of the reasons you have much more of a movie feel than a book feel with orthodox 4e-martial in Middle Earth. I think for the puzzle-monsters you'd have to run one off things anyway and no system would really help or hinder that much as they are all one off.
I think this kind of thing would also make a Doctor Who game hard to do. You probably can't beat the Cybermen... unless you've got gold. You probably can't beat the Slitheen... unless you've got vinegar.
The Dr Who RPG has one briliance in it - the Initiative Sequence. Talk, Act, Run, Fight. (I think - the runners might act before the act (i.e. non-combat actions) people).
4e does a few other things right for LotR too. From what I recall, Gimli took a blow to the head and just kept on ticking. Someone spent a healing surge or two after that fight. But Frodo got stabbed by a Morghul-blade and didn't get better. He was basically suffering from a disease track-like curse, and even Aragorn couldn't actually treat the injury. (He could have used the Delay Affliction, which is 6th-level IIRC, non-core, and then let someone in Rivendell use Remove Affliction on him later.)
This. The Morghul Blade is basically the Condition Track in action - and Healing Surges show up in just about any heroic fantasy; most fantasy heroes spend healing surges at some point or other. That said, the condition track is only a trivial advantage for 4e here as the mechanics are basically a useful bolt-on rule that would work in any game and wouldn't take long to explain.