Hello! And thank you!
Don't run into many Beornings around these parts, I tend to think of him as a lot like Tom Bombadil
I actually haven't read The Hobbit in 15 years or more. The only copy I had was softcover, and my softcover books stayed behind when I moved to Texas 12+ years ago. Luckily, I had hardcover copies of LOTR and The Silmarillion, and I brought my hardcovers with me. Anyway, I think Tolkien wrote The Hobbit at a time when he conceived of Middle Earth as a much more "magical" place than it became later. At least, to me, the idea of a man who can shape-change into a bear seems out of place in LOTR.
As for Bombadil, well... he's a Maiar, isn't he?
Wow... interesting perspective. I look at it differently, I see Hobbit as a fairy-tale and the other stories as more mythic. There are shape-changers in Silmarillion, or at least Elves and heroes who can take on other shapes - like when Beren and Luthien snuck into Angband. Or Elwing, who becomes a bird sometimes to fly to Earendil...
I will say that those days are more distant at LotR - we're more often among "civilized" men rather than wild ones. Rohirrim or Gondorians would probably see Beorn as something out of a child's tale, but he fits into his surroundings. He lives way out in the middle of nowhere, which is where you'd expect to find a relic of the older days.
The one thing I do agree with about your comment is that the magic in Middle-Earth fades. Look at the stories of the First Age; there's magic galore. By the end of the Third Age, it's mostly leftover magics by people like Gandalf or the Elves, and they're leaving. As Men come to the fore, the world becomes more mechanical and less wondrous. Now all we have are the stories...