Tolkein would have had to construct yet another conflict of the sort which drives LoTR, and he has a limited pallet to draw from (Sauron is pretty much KIA, so now what, Morgoth himself, that means portraying a final 'Armageddon' essentially). Basically he'd have to one-up LoTR too, or it would simply be an afterthought. He's already explored the theme pretty systematically, so it may be possible to deepen the examination, but it would not be anything like the previous story. For instance he could actually explore Melkor (Morgoth) in more detail. Is he really the abominable monster he's made out to be in Quenta Silmarillion? Is he even ultimately 'evil' or is his purpose simply to help drive the ultimate plan of Illuvatar? He would have to start to engage with these deeper questions about the nature of good and evil. Why does the omnipotent and omniscient Illuvatar have evil in his plans if he is supposed to be good? etc. It sure wouldn't be the same simplistic (though Tolkein certainly isn't simple in many respects) free will vs tyranny sort of stuff that makes LoTR go!
So, frankly, I am of the opinion that ME, as envisaged in LoTR at least in its 3rd Age incarnation, is pretty much a single-use setting. Yes, you can tell small 'fairy tale' type stories, even up to the scale of "fighting mighty dragon Smaug" or something equivalent, but the story of the War of the Ring is pretty unique, it cannot be rehashed within that setting in an effective way. Given that Tolkien already wrote There and Back Again (The Hobbit) as a prior story of that ilk he's got even less space. At best another similar tale has to equal or best Bilbo's tale, which is going to be hard to do. Even if he pulled off such a thing its hard to give it the salience that was retconned into the finding of the Ring and Gollum, etc.
I mean, various RPG authors have produced material/systems for play in 3rd/4th Age ME. In my experience it doesn't come off as an especially great vehicle for that. The authors are stuck trying to provide some adventures that turn out not to change anything (because they don't have the authority or desire to establish divergent canon). Tolkien didn't actually describe ME in much detail either. Yeah, there's a bunch of place names and various things can be inferred or are casually mentioned in passing, but the setting is actually profoundly deficient in everyday 'stuff'. Outside of the Shire we get very little to no examination of everyday life or ordinary people. When you start to inject that stuff it tends to mar the high fantasy feel of the setting. In the end my conclusion was that something like WoG, my own homebrew, etc. end up playing pretty much equally well in practice and I have a lot more freedom in my homebrew to define the major conflicts and the nature and feel of things.