The word 'beef' entered the English language following the Norman invasion in 1066. It's a word with Latin roots, but came to English via French.
Consider the dichotomy between sheep & mutton, or pig & pork. In all instances, the word referring to the animal has "old English", or Anglo Saxon roots. Likewise, the word referring to the meat is Latin via French.
This linguistic gap may have its roots in social divisions. In the years following the Norman invasion, the upper class spoke French. They also consumed most of the meat.
The conquered, who by and large did a lot of the agricultural work & raised the livestock, spoke Anglo Saxon ("old English").
Regardless of etymology, though, beef doesn't refer to generic 'meat'.
It's funny, In Spanish we don't have an equivalent of "beef", we use the word for meat "carne", but we have the anglicism "bife" (from beef) mostly used in South America, or "bistec" (from beefsteak) mostly used in Central and North America.
"Beeves" is a little archaic, and if you were to poll most ranchers and traders today, they'd probably use "beefs." But it's still acceptable, and it hasn't completely fallen out of usage. (Nor has referring to a cow as a "beef" fallen out of usage, especially in the livestock trade. It's convenient to refer to a cow bred for consumption as a "beef," to differentiate it from a cow bred for dairy.)
"Beef" in the second case is probably short for 'beef cow,' just like 'dairy' is going to be short for 'dairy cow,' not some weird anachronistic usage (unless the Normans once called cows 'dairies.')
Strangely enough, there's actually an anachronistic usage at the heart of it all. The word "beef" came, originally, from the Norman-French "beuf" (precursor of the modern French "boeuf"), from the Latin "bov," which referred to the animal, not the animal's meat.
"Beef" started switching meaning from the animal to the meat because the meat was something only upper-class, Norman-descended Englishmen could afford. They had less experience with cows in the field, and more experience with cows as slices of meat on a plate. So the snootier, upper-crust "beef" came to be used in the context of food, while the Old English, working-class "cow" was still used by the actual farmers in the fields, who were dealing primarily with the animals. Merchants and middlemen, who dealt primarily in the meat but also in the livestock, adopted the more aspirational "beef" to refer to both. This usage was common until as recently as about 100 years ago. There are probably few people around today who'd call a cow "a beef," though there are still people who'd refer to a herd of cows (or any quantity on a market) as a "beef herd," or a "herd of beef."
I'm not going to go out on a limb and claim that "beef" is common, modern usage when referring to the animal. (And "beeves" is now so uncommon as to sound funny to the modern ear.) But the usage is not entirely extinct, and in instances when differentiating beef from dairy, it's moderately useful.
More likely they would say "head", as in "two head of cattle". In all my childhood days on the farm, I don't think I ever heard the word "beefs", much less "beeves". Usage of the latter would likely get you a look as if you had come from back east.
Consider the dichotomy between sheep & mutton, or pig & pork. In all instances, the word referring to the animal has "old English", or Anglo Saxon roots. Likewise, the word referring to the meat is Latin via French.
This linguistic gap may have its roots in social divisions. In the years following the Norman invasion, the upper class spoke French. They also consumed most of the meat.
The conquered, who by and large did a lot of the agricultural work & raised the livestock, spoke Anglo Saxon ("old English").
Regardless of etymology, though, beef doesn't refer to generic 'meat'.