Interesting. I always thought that Tolkien had used this explanation explicitly, perhaps because it was illustrated in David day's Tolkien Bestiary, but in The Silmarillion it says only this (or this is all I can find):
"Of the Sindar"
So it seems he only hinted at it. What's your source for the fact that he didn't like the explanation? Is it mentioned in his letters?
From the Silmarillion;
"Yet this is held true by the wise of Eressëa, that all those of the Quendi [Elves] who came into the hands of Melkor, ere Utumno was broken, were put there in prison, and by slow arts of cruelty were corrupted and enslaved; and thus did Melkor breed the hideous race of the Orcs in envy and mockery of the Elves, of whom they were afterwards the bitterest foes....This it may be was the vilest deed of Melkor, and the most hateful to Ilúvatar."
Tolkien came up with a range of different explanations for the Orcs, and as you'd expect with a complex, constantly evolving project, there are some contradictions there. Here's a summary of his proposed explanations:
1. From the Earth
His earliest explanation is that Melkor made them from the earth, however this explanation quickly became obsolete as he made it clear only Eru Ilúvatar could create life.
2. Avari (East Elves)
According to the Silmarillion Melkor caught some of the Avari and corrupted them into the first Orcs.
3. Sentient Beasts
In "Morgoth's Ring" he proposes that Orcs are essentially beasts, corrupted into humanoid form, which have only the mimickery of speech, like a parrot. I would note, however, that this directly contradicts how Orcs are portrayed in "The Hobbit" and "The Lord of the Rings" where they are clearly intelligent and have free and independent will, even forming social order.
4. Fallen Maiar
The "History of Middle-earth" series has passages that strongly suggest that some particularly notable Orcs are actually fallen Maiar.
5. Corrupted Men
According to Christopher Tolkien in "Morgoth's Ring", at some point Tolkien began to reconsider the "corrupted from Elves" explanation, and instead started looking at them being created by the corruption of men by Morgoth. However this would have required a major modification of the timeline as men hadn't been awakened when Orcs first appeared. Tolkien died before he could make this change.
6. Corrupted Men and Elves
A later explanation, possibly to address the problem with the above explanation, is that Orcs were a combination of corrupted Elves and corrupted Men.
Again, from "Morgoth's Ring" (note the later spelling change to "Ork")
"Since Melkor could not 'create' an independent species, but had immense powers of corruption and distortion of those that came into his power, it is probable that these Orks had a mixed origin. Most of them plainly (and biologically) were corruptions of Elves (and probably later also of Men). But always among them (as special servants and spies of Melkor, and as leaders) there must have been numerous corrupted minor spirits who assumed similar bodily shapes. (These would exhibit terrifying and demonic characters.)"
Finally, another suggestion in "Morgoth's Ring" is that Saruman later cross-bred Orcs with Men, possibly explaining the creation of Saruman's Uruk-Hai. (Treebeard also suggests this in LOTR itself).
Dismissing the first explanation (as it directly contradicts a fundamental principal of the world), I see no reason why the other explanations can't
all be true. If we take it that the Orcs were
originally created by corrupting Elves, this doesn't exclude Melkor/Morgoth and Sauron later using the same techniques to corrupt Men and beasts to create additional Orcs (presumably Men and beasts would be easier to corrupt, explaining why later efforts would be focused on them). Neither do any of these explanations preclude individual "special" Orcs being corrupted and fallen Maiar.
Finally, Saruman explored cross-breeding of his existing Orcs with Men to create Uruk-Hai.
