Klingon, Dothraki, the belter language in "The expanse" - all tv shows try hard to create something new.
And I dont think politics is the reason here.
(TV shows are a totally different thing, of course. They definitely try to create something new, but that’s because they’re TV shows, not because they don’t like other languages. It’s the same reason they’re set in outer space instead of being set on Earth.)
Many of these were influenced by real languages: Sindarin by Welsh, Quenya by Finnish, and Dwarvish by (I think) Semitic languages. As you pointed out, he knew a lot about linguistics.
As for Orkish: I'm sure Tolkien makes a point of saying orcs had no language of their own; rather, they used a particularly crude form of the Common Speech, often peppered with words from the Black Speech. (This is the in-story reason why the protagonists in The Hobbit and The Lord of the Rings are able to understand and be understood by goblins and orcs.)
Anyway, none of this detracts from your point: invention of fictional languages does not imply a reluctance to get to grips with actual non-English languages, at least not in the case of J.R.R. Tolkien.
/end pedantry
As an aside, I don’t think we could call his rough sketch of dwarvish as “full”. AFAIK we only know a couple of words of it. He did develop 2 elvish languages quite extensively, though (Quenya and Sindarin), and worked on several other dialects.
He created Quenya and Sindarin as more or less complete languages, with detailed grammar, extensive vocabulary and several writing systems. They are both Elvish languages, descending from "proto-Elvish", so he carefully tracked their evolutions, sound changes. The other Elvish languages and dialects were mentioned and/or alluded to, but not developed in detail.
Next, there is a series of languages he has developed to a far smaller extent than the previous two. The best known examples are:
1. Khuzdul, or Dwarvish, which exists as only a few words and phrases, spoken and/or written in his works, plus some general grammar rules. The explanation for it not being more known is that the Dwarves were keeping it secret, and used other languages for talking to any non-Dwarves.
2. Black Speech, used by Orcs and other Sauron's minions. Developed by Sauron and also only present in a few sentences and words -- as well as, of course, the Ring inscription.
3. Adunaic, or Numenorean - the most detailed described human language, similar to Khuzdul in some grammar aspects.
And finally, he has used real world languages to "represent" Middle-earth languages. The best examples are English representing Westron - the lingua franca of the western Middle-earth - and Anglo-Saxon representing language of the Rohirrim, to show its relation to Westron. That said, in later years ye has started "developing" those languages more, inventing e.g. Westron words to replace the English ones used in the books (e.g. "Suza" for "Shire").
And Belter is meant to be an English-based creole, reflecting the in-book/in-show history of the Belters as the descendants of speakers of several different languages who needed a lingua franca.
Would it matter? None of the characters are supposed to speak English in-universe either.
But it might jar to hear a group of aliens natively speaking Dutch, for example. (Unless it's a Dutch film for a Dutch audience, of course, in which case it might equally jar to hear them speaking English.)
> people with different accents
Done to (arguably) comic effect in the BBC sitcom 'Allo 'Allo!, about the French Resistance in WW2, where English, French and German are all indicated by outrageous caricatures of the relevant accents.
Heinlein did something similar with the Luna patois in The Moon is a Harsh Mistress.
I'm sure it's been done many times, both in literature and in fact.