There has been a significant effort to eradicate coyotes. Every year, about half a million coyotes are culled in the United States. About 60-80,000 are killed by the federal government. Trust me, there's an effort. Coyotes are favorite targets of "killing contests" in which basically the goal is to bring in the biggest coyote. Sometimes, coyote whacking occurs, a horrible sport in which coyotes are run to exhaustion with snowmobiles and often run over multiple times until they're dead. In my experience, while wolves have recently gotten better public support (people consider them beautiful, powerful, mystical, etc.), coyotes remain hated, considered pests, and this is only occurring more now that coyotes are becoming more urbanized.
As a note to that, most folks who kill coyotes don't realize that killing often makes the situation "worse". Usually, coyote litters have a 25-30% survival rate when under low mortality pressure from humans. But higher human mortality causes an instinctual "population buffer boost" that makes litters increase in size and shoot up to 60-90% survival rate.
Wolves keep coyote populations in check; they're territorial and don't tolerate encroachers. When wolves are around, coyotes generally are suppressed by 20-50%. So when gray wolves were eradicated from the lower 48 by the late 1920s, coyotes literally became top dog. Their numbers boomed, and moved east. As they moved east, they mixed with some remaining wolves whose populations were so low that they considered coyotes possible mates, not enemies. That's why eastern coyotes are often larger than western coyotes, and where the word "coywolf" came from.
Coyotes do numerically take more livestock than wolves, but you have to consider how many more coyotes there are than wolves. In the lower 48, there are about 5,000-6,000 gray + eastern wolves, and <25 wild red wolves in NC. In NC alone, my state, there are an estimated 50,000 coyotes AT LEAST. And that's just 1 state out of the 49 they currently occupy. So per capita, coyotes don't necessarily take more livestock than wolves. Coyotes also rarely take adult cattle, usually if there's more than one and if the cattle is sick. They prefer easier targets like small sheep, lambs, and calves.
Source: I'm a carnivore biologist doing field research on eastern coyotes/coywolves + statistical research on gray wolves. Hope this helps!