A win-win situation, having in mind the fantastic work done from those cats on protecting orchards, timber plantations and fruit trees. Benefits that every rural-apocalypse prophet will carefully omit to mention by ignorance or greed (and that in fact may bring more money to the community that the lamb lost). If somebody works for your benefit, the right thing is to paid them somehow.
What it matters in economy is the final balance among loses and benefits. This people just are blind and fail to understand the concept each single time.
I'm not sure if you're just venting or genuinly curious, but the economy doesn't add up. Raising 50 extra animals would bankrupt those farmers you're talking about.
European lynxes eat basically solitary roe deer and snow hares (In that sense they benefit, not harm, sheep farmers). They rely strongly on stealth and probably would have a hard time ambushing sheep from a flock with thousands of eyes watching. When discovered at time by their preys, lynxes abort the chase.
Extreme right and populist parties growing in Europe are unfortunately pushing and pushing the opposite narrative with the goal to milk grants from government, but is based in fairy tales not scientific facts and in the end this plot will simply cost a lot of money to everybody, (as usually each time science is ignored).
Iberian Lynxes are like half of European ones on size. Evolved to eat basically rabbits, so they protect agriculture.
Their main conflict could be with chicken owners. In the few cases where young cats without a territory were desperate enough to break in human buildings, farmers were given one-time money to fix their coops and advice to made it lynx-proof. The animal is a beauty and most people don't care really, specially when they realized that if there are lynxes around, the ever-trying foxes vanish (Lynxes kill foxes at sight and avoid people so the result is a drastic reduction in the yearly number of coop attacks). Adults don't need it and they go for the rabbit of the day that is much easier to catch.
As someone who has been raised in a rural area next to Iberian Lynx, I can confirm they wouldn't dare to kill anything larger than rabbits.
As for chickens, they tend to stay away from human populations. Foxes and genets are most likely the ones to prey them. And like you've said, thriving Lynx populations keep foxes at bay.
Either way, other than some killings made by wolves (those do make a dent on a herder's living) and wild boars just making a mess, no one in my hometown was ever stressed by animals doing their thing.
If anything, they were relieved mother nature was healthier than they had thought. I often see that it's important to say this part out loud: those in rural areas care about nature. Hunters included.
