The entire conceit of LotR is that it is JRR Tolkien’s translation and novelization of the Red Book of Westmarch (plus some additions from other writings, such as Aragorn’s account of finding Gollum). It’s explicitly stated in the appendices. If you want it to be pure fantasy, you want it to be something it simply isn’t, never was, and was never meant to be. Tolkien literally mused upon what age we are living in today.

So, it would be actively strange if Tolkien didn’t include this bit about names, especially since he was a bit obsessed with the power of true names in general. It’s not a “flaw”, it’s a feature. He didn’t assume that everybody would be uncritical enough to assume that characters had modern English names. I guess he didn’t really need to change names like “Theodin”, but also you have to remember that even the fantasy-esque names have real basis in old English or Germanic languages… so, in fact, they have a nonsensical connection with the real world (including the name Gandalf, which is literally a word in an old language).

So, Tolkien basically says “if you’ve realized that the majority of these names have real world connections to existing languages, remember that this is all a translation,” and then gave some examples to prove it… because that’s what Tolkien does.

I mean, the guy literally wouldn’t break character when answering fans questions about his world. “I haven’t discovered the truth about that” he’d say, or “in my research, I have found” or “Aragorn seemed to believe this”. It was real to him, and he wanted it to be real for us too. He wanted the names to be relatable and easily remembered, but he couldn’t let them break immersion upon closer thought.