I suspect the main reason why flesh is not commonly associated with animals' carcasses stems from a sense of aversion, and sanctity.
Consider the expression, flesh and blood, which is used in connection with our children, (‘She is my own flesh and blood’) our family, and our fellow beings. “We are only flesh and blood”, means we are only human, we are imperfect; but until very recently, Christians were taught that man was made in the image of God. This allowed them a greater status than the animals that roamed freely on land or in air. Our flesh deserved greater staus than that of mere beasts.
It was necessary to remind Christians that Man was unique; distinct from any land, sea or air creature.
All flesh is not the same flesh: but there is one kind of flesh of men, another flesh of beasts, another of fishes, and another of birds.
King James Bible: 1 Corinthians 15:39
Furthermore, Eve was made from the flesh of Adam:
And Adam said, This is now bone of my bones, and flesh of my flesh: she shall be called Woman, because she was taken out of Man.
King James Bible: Genesis 2:23
And when a couple married the expression one flesh was used to convey the intimate union between two human beings.
And they twain shall be one flesh: so then they are no more twain, but one flesh.
KJB: Mark 10:8
In the New Testament, The King James Version contains the above mentioned idiom
And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
In Jeremiah 19:9 the following horrific revelation was made
And I will cause them to eat the flesh of their sons and the flesh of their daughters, and they shall eat every one the flesh of his friend in the siege and straitness, wherewith their enemies, and they that seek their lives, shall straiten them.
The horror is all the more heightened because to eat the flesh of a dead human child is not comparable to the meat of a dead cow, sheep, or pig.
The distinction between flesh and meat has been instilled in us for over six hundred years. If we look at the etymology of ‘meat’ we find
Old English mete "food, item of food" (paired with drink), from Proto-Germanic *mati (cognates: Old Frisian mete, Old Saxon meti, Old Norse matr, Old High German maz, Gothic mats "food," Middle Dutch, Dutch metworst, German Mettwurst "type of sausage"), from PIE mad-i-, from root *mad- "moist, wet," also with reference to food qualities, (cognates: Sanskrit medas- "fat" (n.), Old Irish mat "pig;").
Narrower sense of "flesh used as food" is first attested c. 1300; similar sense evolution in French viande "meat," originally "food." In Middle English, vegetables still could be called grene-mete (15c.) […]
Colin Fine in the comments asks:
But three hundred years ago, there wasn't a word in English which meant what meat means now: the only word for "animal's flesh for eating" was "flesh", So how can you claim that the distinction between meat and flesh goes back two thousand years?
I admit, I didn't know the term flesh was older than meat; the former is before 900 while meat meaning ‘flesh that we eat’ is from 1300; and that meaning is older than three hundred years.
It is safe to say that by Early modern English, the distinction had been established. The above quotations are from the King James Bible, first printed in 1611-1612, and I believe the division between ‘human flesh’ meaning soft human tissue and ‘meat’ was reinforced by those renowned passages.
The OP claims that English uses the term flesh in compound words referring only to certain animals, e.g. horse flesh, but not with other domestic or equally as intelligent animals such as dogs or dolphins. If we look at the Ngram of the British English corpus, which is arguably older than its American English counterpart; we see that the compounds horseflesh and its hyphenated variant, horse-flesh, were the far more common than horsemeat (blue line) or horse-meat.
“It seems, however, that as recently as 1817, there was a quantity of horse-flesh sold for human food in Paris” (Essays on Physiology and Hygiene, 1838)
“Many authors are quoted with regard to the wholesomeness of horse-flesh, whose opinions differ. One says, "that of those who eat the flesh of diseased horses, nine out often die; it should be roasted and eaten with ginger and pork."”
(The Chinese Repository, 1839)
“... and, divesting the mind of the inferiority of horse-flesh over cow or bullock-flesh, the food of hounds, both in its nature and cooking of it, is such as man might not only reject, if necessity compelled him to have recourse to it, […] It is a common expression, that “any thing will do for the dogs,” but hounds, to be in condition, must have every thing good of its kind, and also well cooked. [...] and well-boiled horse-flesh, quite free from taint. [...] When taken out of the boiler, it forms a substance resembling coarse rice pudding; and when the fresh flesh, which is shredded, and the broth in which it is boiled, are added to it in the trough, and very well mixed, it forms the best and highest food that can be given to hounds.”
(The Horse and The Hound, 1842)
“Since the siege of Copenhagen in 1807, horse flesh has regularly been sold by the butchers in that capital for general consumption.”
(Monthly Retrospect of the Medical Sciences, 1848)
Overwhelmingly, the meaning of horseflesh (the equivalent of today's horse-meat, and horse-beef) is that of food for human or animal consumption.
The shift from horseflesh meaning meat=food, to the live animal's flesh began after the 1970s, and it appears its popularity as a source of food has been decreasing ever since.
“She really was a slick little piece of horse flesh except for her occasional kicking spell.” […] “He had one mare, a splendid piece of horse flesh, who had raised him twenty-seven head, besides giving him years of hard work.”
(Mister, You Got Yourself a Horse: Tales of Old-Time Horse Trading, 1981)
“At that time I was not a very good judge of horseflesh. The horse appeared sound and gentle, and, as the owner assured me, had no bad habits. The man wanted a large price for the horse, but finally agreed to accept a much smaller sum, ...”
(The Conjure Woman, and Other Conjure Tales)
Good Judge of Horseflesh:
”A man with good taste in women. A buyer, trader, breeder, or racer of horses who is a good judge of horseflesh is able to size up the potential value of a horse just by looking at it or watching it perform. A bachelor is a good judge of horseflesh if he consistently escorts beautiful women or manages to convince the most beautiful one to be his bride. Granted, the woman is being equated with horseflesh; but at the wedding, it is the man who becomes the groom.”
(Speaking of Animals: A Dictionary of Animal Metaphors, 1995)
Finally, one long excerpt from a book entitled Unmentionable Cuisine
If asked to explain their personal aversions to eating horsemeat, a few Americans might say something about the taste of the meat not comparing favorably with beef. Many others would mention the “nobility” of the horse and therefore its unsuitability for food. Perhaps a few people would relate their horsemeat prejudice to the lingering effects of prohibitions by the Catholic Church that were intended to break the horsemeat-eating habits originally associated with horse worship by the ancestors of many of us.
In pre-Christian times, horsemeat eating in northern Europe figured prominently in Teutonic religious ceremonies, particularly those associated with the worship of the god Odin. So much so, in fact, that in A.D. 732 Pope Gregory III began a concerted effort to stop this pagan practice, […] The Angles of England were among those peoples who regarded the horse too holy an animal to eat routinely, reserving it for communion meals, and some believe that this prohibition has carried over into the strong prejudices in England today against eating horsemeat.
Horsemeat is called chevaline in France and often, like pork, is sold in separate butcher shops. From time to time there has have been strong movements to increase the use of horsemeat in French cuisine, for it can be prepared essentially as one would beef. […] during the beef shortage in 1973 and because of high beef prices since, quite a bit of newspaper publicity has been given to butcher shops doing a land-office business in horsemeat […] In 1972 the U.S. federal veterinary services passed the meat of 68,000 horses as suitable for human consumption.
Unmentionable Cuisine By Calvin W. Schwabe (1979, and fourth printing 1996)
Note, that not once does the author use the compound horseflesh in the entire chapter, horsemeat refers to the carcass of the animal in question.