Nope, it’s completely fine to eat cow or horse meat cooked like that (but not ovines, pigs or birds). It’s very common in Italy. Never heard of food poisoning from this kind of meat. I do hear of food poisoning from chicken meat or seafood, though, as they are more prone to host dangerous parasites. I would still carefully select the meat vendor or restaurant, though.
I’m not a vegetarian, but I consume very low amount of mean compared to the average (I’d say I consume no more than I need), and I totally agree that there should be no difference between cows and horses, as both are very intelligent and can get attached to humans or other species. Same goes for many other mammals, and not limited to.