Why don’t goats produce wool the way sheep do?

Strictly speaking?

Because goats produce their own spinnable fibre – notably the Cashmere goat breeds and the Angora goat, which produce cashmere and mohair respectively. Yes, mohair – not angora, which is produced by the Angora rabbit.

Historically, or rather paleoagriculturally, sheep appear to have been bred primarily for their wool, whereas goats seem to have been bred primarily for milk, although there are such things as milk sheep and, clearly, fibre goats. I can only hazard a guess that the primitive sheep domesticated by our ancestors must have produced more fibre than goats did*, and/or that primitive domesticated goats had more milk or were easier to milk than sheep

Milk goat

Milk sheep.

Yep, definitely easier to milk the goat.

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* – The Jacob sheep, a relatively primitive breed, produces about 2–3kg fleece, whereas (modern) Cashmere goats produce only 1kg.

Quora linky.