Thanks for checking that out, Jim.
I thought about whether to include Ents in my last post, but as you say, that's a too complicated question (ditto dragons).
However, your mention of Treebeard's poem/dictionary of creatures, and the Ring incantation, got me thinking. You quoted the former directly, the latter goes like this:
Three Rings for the Elf-kings (*cough**cough*), under the sky
Seven for the Dwarf-lords, in their halls of stone
Nine for Mortal Men, doomed to die
etc.
Perhaps the impression I discussed in my earlier post comes from the way "mortal" is so often attached specifically to humans in the (in-universe) lore of Middle-earth. Beren the mortal man is contrasted with Luthien the immortal elf. In Treebeard's poem, humans are contrasted from other creatures because of their mortality, and their mastery of horses (which sounds like a more Rohirrim thing than a human thing, and aren't elves also adept with horses?). And then the Ring poem.
In the second and third cases, humans are especially singled out for their mortality, even though both include dwarves, who also are "doomed to die" eventually from age or disease or somesuch if not by violence.
So it does seem that even though "mortal" can apply to both humans and dwarves and perhaps other creatures, mortality is specifically attached to humans (and, by implication, Hobbits as a human sub-race).
Why might that be? I can think of one potential explanation, though it is only a guess.
Perhaps "mortal" does not simply mean "one who eventually dies of natural causes." Perhaps it has a more targeted, colloquial meaning "one who eventually dies of natural causes
and passes out of Arda forever when they die." Dwarves do not leave Arda when they die, they go to Durin's Halls to await the Dagor Dagorroth. Only humans and Hobbits (and Luthien and Arwen) truly leave Arda and go to join Illuvatar upon dying. Also, there are implications that humans are special because Christ will take their form and walk among them when he comes into the world.
Again, just speculation on my part. I feel the evidence is suggestive, but it's by no means conclusive. Still, it's one possible explanation of why the word "mortal" and synonyms for mortality continually pop up in the thumbnail descriptions of humans as opposed to the other races.