I actually dislike old westerns because of the intentional cruelty and wasteful damage to horses in old films before oversight by humane organizations.... But I think for the most part, treatment and methods have improved, to the benefit of both the animals and the films.
But it'd be silly to pretend that it isn't still dangerous, and those overseeing the treatment of the animals on set have to always balance that. Good horsemanship can make it saf-er.... never pretend it will ever be completely safe.
There was a lot more risk of harm both to people and animals, because nobody had time to be super-careful. More importantly, horses weren't handled exclusively by folks who like and care for horses, but just by anybody who needed to go someplace. Especially in the case of cowboys, who might need to catch and saddle 4-5 half-broke cow ponies just for their own transport in the course of a day. Just think how some people treat their cars, and you'll have an idea of how folks in the old days treated their horses - unless they had a valuable horse or were particularly humane people.
A lot of old time cowboys left a lead on horses in the remuda because it made 'em easier to catch in a hurry. And if the horse broke its neck, well, it was just a horse. They weren't valued or trained as intensely as we train our horses today - unless it was a special cutting or roping horse like Bill Pickett's Spradley (in which case it wouldn't be loose in the remuda in the first place).
The cowboys are going to be the very last ones to get with the "new thinking" - because they are still working with unbroke and half broke mustangs with little if any monetary value except for dog food.