My thinking is that the Eagles of Manwe are more than mere birds, the way Ents are more than mere trees, and fully incarnate. It isn't weird to me that they can speak. The Gaurhoth, Wargs, Vampires, and Demon-lions are demons and their spawn, that isn't weird to me either. Ungoliante is some sort of demon I guess?, but other than maybe Shelob I don't see evidence of other giant spiders speaking. (Can they even understand Bilbo's Westron? Sometimes I wonder if the Ring translates things for its wearer. He seems to understand Nandorin and Orkish without trouble.)
Huan is a special case even compared to the rest of his kind, apparently. The Thrush can't speak any human language, and I get the feeling those black thrushes are an unusual species (they're a fictional one, as far as I know). The ravens can speak in sentences, but real ravens can mimic words at the least. So, for creatures that really deserve to be called animals, there isn't much speaking. The fact that Elves and some Men can understand the "speech" of birds and beasts, and Beorn can talk to any animal, makes me wonder how intelligent (sapient??) the average Middle-earth animals are, but mostly they behave just like real animals so we don't have to worry about it much, I think. There's no need for talking foxes, is what I mean.
Likewise, when Legolas says he can hear the speech of grass and rocks, I don't think he means that literally 😉.
Gurthang is just totally weird and I have no idea what to say. I assume that Bilbo embellished his tale with silly things like talking purses and 11-foot-tall human!Beorn, and it seems Tolkien embellished it further with English anachronisms. But Gurthang is just... I don't know. Did that even really happen, or was it inserted by the Narn author Dirhaval? Who (alive) was on hand, anyway, to hear Turin's last words? Does Mablung have keen enough hearing to hear them from some distance? Not saying we should need to get rid of that, but I don't know how to depict it so it isn't weird. At the least, there is the line by Gwindor saying the sword mourns for Beleg.