The dogs we call “greyhounds” today are smooth-coated. Some, like Drake, grow pretty thick winter coats, but they are all smooth-coated dogs.
This breed feature, one would think, would have been well-established in the breed, but it did not become a feature until the rise of “club coursing” in England. English patricians were part of clubs that had vast holdings where hares were managed to abundance, and each club member would come and run dogs on these hares in what were greyhound field trials.
In his book Greyhound Nation: A Coevolutionary History of England, 1200-1900, Edmund Russell writes about how memes affected greyhound evolution in England. One of these memes that arose in the 1820s was that no rough-coated greyhound could be entered. Russell even quotes the rule-book from the Swaffham that “No rough-haired dog to be deemed a greyhound.”
The reason for this dismissal of rough-coated greyhounds is that rough-coated greyhounds were better suited to the North of England and to Scotland and Ireland. The wire coats protected against thorns, rocky terrain, and the cold weather. Smooth-coated dogs were better for the open land, which was more easily found in the South of England.
And it also fit within the general prejudicial views of the elite society of the South of England, which saw itself as being “better-bred” than the North. This division is one that sort of posits the Anglo-Norman parts of the country against the areas where Celtic people held on the longest. This same view was even more exaggerated when these elites looked at the traditionally even more Celtic lands of Scotland and Ireland, where rough-coated greyhounds were the rule, not the exception.
Further, Russell points out that was about this time that lurchers began to be stigmatized among elite coursing circles. The lurcher was seen as the poacher’s dog, and the poacher’s dog very often was a rough-coated creature. Never mind that the intellectual ancestors of these elite coursing men were very much into the business of crossing greyhounds with lurchers, Italian greyhounds, and bulldogs. The lurcher and the wire-coated greyhound began to be seen as debased and low-class and Celtic.
One should also take into account the coursing men were never running hares for food. This was sport. The lurchermen was always running dogs on what could feed him and his family the next day. The wealthy coursing men were interested in the dogs as sport, not survival, and for this difference, they castigated the rough greyhound as a lurcher too.
So these ideas permeated what became the modern greyhound breed. Wire coats are dominant, and it is quite easily to expel them from a breeding program.
Thus, what became the greyhound resulted from the prejudices of the elites in the South of England and Yorkshire, rather than any practical reason.
In the Scottish Highlands, the “deer greyhounds” of that region remained wire-coated, and they eventually became the basis of the modern Scottish deerhound breed. By the time these dogs became part of the modern kennel club registry system, no one really thought of them as being related, and no one really considered the bizarre class and regional reasons why the two breeds wound up with different coats.
The deerhound retained the wire coat because it was practical for running deer in the Highlands. The greyhound lost its wire variety because the elite who coursed them saw them as a sign of debasement and excluded them from the breed.
Human whims and prejudices affect so much of how our breeds evolve. In this case, it is the deep division of the South of England vs. the rest of the British Isles.
Update: Apparently, when a real historian looks at the history of a dog breed within the context of the culture and with a clear understanding of canine genetics, it upsets the apple cart with the Sir Bufton Tufton Brexit and Boris set. I don’t have have time for to approve your comments and show you why the lore is not the same as actual history.
(And here is the genome-wide study that has a phylogenetic wheel in it that places Western sighthounds as close to British herding dogs.)