A few general comments:
1) D&D was indeed emulating Tolkien, but it's not like interbreeding is unique to Tolkien. It is extremely common in myth and fantasy for elves, dwarves, giants, titans, gods, demons, monsters, and others to interbreed.
2) Having two defined half-races (elf/human and orc/human) was clearly a nod to Tolkien, but a very limited one that doesn't even cover the main characters such as Aragorn - whose fractional elvish ancestry is clearly very significant in the books.
3) The simplest approach to interbreeding is to allow racial characteristics to be chosen to match the character's breeding, rather than having static packages for all possible combinations. It has a lot of parallels to class-based vs. skill-based design for profession/training. This could be easy, say, in Fantasy Hero, GURPS, or other broad point-based systems (FATE, etc.). There's nothing wrong with fixed race packages - but if one wants to have a setting where different fantasy races breed true (which is the topic from the OP), then this seems like the best way.
4) In D&D, this would obviously take changing the rules. The result would probably be something more like a point system. There have been a few of these for D&D, like those to allow playing arbitrary monsters, for example.
I give the player three options if they want to play a hybrid that's not in the books (half-elf, half-orc, half-ogre*):
1) If a human + demi-human/humanoid hybrid is similar to one already in the game, they can use that template. For example, a half-dryad would be a half-elf minus the pointy ears. A half-hobgoblin will simply be a taller half-orc with maybe greyish skin.
2) Hybrids of other demi-humans have to choose one parent or the other. A dwarf/halfling can either be a taller halfling with a thick beard, or a beardless dwarf with hairy feet.
3) All other human, humanoid and demi-human crossbreeds are considered human, maybe with some vestigial trait from a non-human parent. A pixie and a hill giant get it on? They give birth to a human kid.
I handle it this way for a few reasons. One, I'm not about to fabricate stats for every possible hybrid. Players can adapt an existing one, flip a coin and choose a parent for the PC to emulate, or just be a human with a couple of odd branches on the family tree. Two, as you point out, myths and fairy tales are chock full of stories where gods, giants, ogres, titans, fairies, dwarves, nymphs, dryads and other fanciful anthropoids bang away with humans and produce offspring -and in most cases that offspring is human. Even the offspring of gods and nymphs will be human after a generation or two.
I think of all these human-like beings as analogous to different types of wild or domesticated canines. There are a dozen or so types of wolves, jackals, coyotes and wild dogs. While at first they might seem very different, in fact their DNA is almost identical and they will readily interbreed with each other (and domestic dogs), producing fertile offspring. After a few generations, these mixed-breed dogs start resembling something like the dingo or pariah dog or tropical wolves, only with some signs of mingling between types. For example, in some populations of gray wolf there are some with solid black fur. This turns out to be the result of breeding with domestic dogs, as are piebald patterns that pop up in dingoes, jackals and other canids. As George Carlin used to say, this is the kind of thing I think about when I'm home alone at night during a storm and the power goes out.
In other words, all the humanoids are basically humans that have been molded by some combination of evolution, magic or selective breeding by more powerful beings.
* I added the half-ogre from Dragon Magazine.