The primary useful thing about a sheep is not its meat. It’s the wool, which is a renewable resource every year. You can make the wool into useful items; you can sell the wool and buy other stuff. You can breed the sheep and have it make more sheep, at the same time it’s making you wool. Its lanolin on the wool also soothes your hands and helps cure wounds. And you don’t even have to pay for the feed; the sheep eats grass that seeds itself for free! Finally, when the sheep gets too old for any other use, you can eat its meat.
If you kill a lamb for Passover, you are sacrificing its entire useful lifetime of producing stuff for you. You get to eat the meat, yes, but that’s only one meal. Basically, you are destroying one of your little factories so you can eat the machinery.
Turtledoves and pigeons also are useful for more than just meat, although not as useful as sheep.