I'll be the contrarian.
The BO I rode for insisted that all horses on his place wear halters 24/7. He claims never to have had a problem with it. In the 10 or so years I was there, I never saw a horse get hung up. Some horses seemed to have had a knack for losing their halters, but I never saw or heard of a problem due to having a halter on in the pasture.
When I was a teen, I saw a horse hung up in the crotch of a forked tree, dead. The horse had no halter. We think it was chased by a dog, but no on actually saw it happen. Horses can get into big trouble with or without a halter.
When I was a kid, our horses sometimes had halters, sometimes not. Usually not. But they came running when we called. As
explained, our call equaled food.
The horse I own now spent the first 3 years of her life on a picket rope, as does nearly all large livestock here. We kept her on the picket rope for several months, mostly for security, and to get accustomed to, and associate us with food. When we turned her into the big pasture, we let her drag the picket rope for a while. Now, she has no halter, but comes to investigate when she sees us or hears our call.