Pheets is right: To Err is Human. And unfortunately, the management you complain to isn’t typically the person feeding your horse. The people throwing hay are probably the least paid and therefor the least careful. It’s not an excuse, I know. Complaining isn’t always an option, because it can tarnish your presence in the barn and you don’t want to make people spiteful of you. You also risk decreased care to your animal by the ‘burned’ staff in the future. They may ‘accidentally’ leave more twine in your hay or ‘accidentally’ forget to top off your water. It’s wrong, but it happens.
Thankfully, (most) horses are rather smart and know not to eat twine. Then again, my mom’s old QH gelding would eat anything within a 3 foot radius of his head, including the occasional plastic bag and sponge. I think for your horse, if anything, he’d get it in his mouth, think ‘yuck’, drop it, and move on to tastier options. I’ve encountered everything in my feed bin, from rocks to dead snakes to rusty barbed wire to *gasp* mold (and the vet bill that ensued). My horse is boarded on a military installation so the hay quality isn’t always great (or good, even… it’s the government, after all). Complaining to the manager got me nothing but a headache and a lot of being told how much worse it could be. Sigh. Right.
At the end of the day your horse is your responsibility, no matter where you keep him. He is lucky to have you checking in