I was not an English major and can't address critical theory issues. I did, however, take two courses in political theory during college, and I don't see where Moorcock's comments in that article have much to do with it, unless you consider them a sloppy way of piggybacking on Marx. Honestly, it seems that he feels an aversion to Tolkien's work, and therefore has come up with criticisms in response. (While this is only my personal impression, I think that many Tolkien haters suffer from a visceral gut-reaction to which they have attached arguments. I also think that most of them hated The Lord of the Rings so much that they never bothered to read The Silmarillion, which offers situations that are far more morally complicated than those that exist in the LotR. Evil Elves! Imagine! Sadly, most Tolkien haters don't.)
From the article, I gathered that Moorcock's slamming of the "There and Back Again" theme was aimed directly at The Hobbit, in which case it is somewhat defensible--if you focus only on that book, and not at Bilbo's larger life-story, let alone the full history of the War of the Ring. The problem with slamming the "There and Back Again" theme is twofold. First, the story is a children's book. If I recall correctly, Tolkien eventually wished he had written the book at the same level as The Lord of the Rings. So it important to recognize that Tolkien grew quite a bit as a writer and mythologist after The Hobbit was written. Second, Bilbo does change. That's the whole point of the story. He starts out as a person who is growing into a rather dull middle age with a narrow, self-centered view of the world. His adventure forces him to change. Perhaps the change isn't as large a change as Moorcock would like to have seen, but that's Moorcock's personal preference. It doesn't necessarily mean that anything is wrong with the story as such. And of course, if you look at Bilbo's life story, the Ring turns out to be evil, and even though he never bears the brunt of it, he does eventually become restless, leave the Shire, and ultimately abandon Middle-Earth altogether.
As to the charge that Tolkien offers “a pernicious confirmation of the values of a morally bankrupt middle class.” First of all, I think it is a good rule to become suspicious any time someone starts talking about a "morally bankrupt middle class." That's old cant--I know it goes back to Marx, and my guess is that an Enlightenment philosopher or two said something similar before him. Possibly also Rousseau, in attitude, if not explicitly. At any rate, by now that rhetoric doesn't really mean anything, especially considering that Moorcock could probably be considered middle class himself. (And his rebellion is, in many ways, a typical rebellion of the middle class against itself! It's not uncommon to find people who benefitted from growing up in the middle class insulting the middle class.) In any case, Tolkien would have been the first person to have agreed that the middle class was morally bankrupt. Tolkien was a very traditional Roman Catholic, and he believed that Original Sin applies to everyone--rich, poor, and middle class all included. To top off his religious beliefs, he had a fairly pessimistic temperament, especially later in life. To consider Tolkien's vision a cozy conservative one is to misunderstand who he was as a person.
Hobbit values (unlike those of other races) do align somewhat with a rural English middle class, and Tolkien makes it very clear that Hobbitry has both a good and a bad side. On one hand, Hobbits are simple, peaceful creatures, and those qualities are admirable. Making things more complicated is not necessarily a virtue (a view Moorcock obviously disagrees with in his "Epic Pooh" essay--he apparently believes that superior writing is always characterized by tension and moral complexity). On the other hand, Hobbits can be narrowminded, foolish, and maddening. For a Hobbit to be very helpful to the rest of the world, he has to be pushed out of his comfort zone, as many of Tolkien's Hobbits are. Bilbo, Frodo, Merry, and Pippin all have ties to the more adventurous Hobbit families. According to Tolkien, Sam is the normal Hobbit, and Tolkien knew that some of his readers who had liked the book found Sam irritating. Sam's harsh behavior toward Gollum is one reason for Gollum's betrayal, but it was normal Hobbit behavior--Sam simply wasn't a broad enough thinker to realize the possible consequences of his actions until afterward. Tolkien had served with less well-educated men during WWI, and he used his stories to show their virtues while exploring their weaknesses.
As I said, Tolkien was staunchly Catholic, so, yes, Christianity lies at the back of his writings. He was conservative, but not in a way that would neatly fit into any current definition of the term. (He noted in one of his letters that, where government was concerned, he favored either anarchism or an unconstitutional monarchy. Basically, however, he was apolitical. None of his published letters deal very much with the British government.) He was also an environmentalist before environmentalism existed as a movement. Yes, he partly wanted to preserve the environment because of nostalgia for his childhood. He also simply loved trees. And, like many of his generation, he was a WWI veteran concerned about the effects of total war. Mechanization worried Tolkien because people were implementing inventions that destroyed the landscape and/or killed other people. The dropping of the atomic bomb horrified him. Tolkien also opposed imperialism and unbridled capitalism. Conservative? Sure. But most so-called "conservatives" today, at least in the United States, would hate the man's guts. The better you know Tolkien as a person, the more difficult he is to pigeonhole.