Non historian--
1, no sunscreen, so yeah as hot as it is, protecting yourself is better.
2, protection from brush, bugs and everything else. Yeah you want cover, even in the sun. Long distance riding a horse in shorts is a no go.
And when it isn't summer, you don't have much "inside " to get out of the weather hot or cold.
So yeah.
Just weighing in here with a suggestion toward comparative research: please refer to any number of historical civilizations that thrived in similar/worse conditions (deserts, etc.), and still do to this day. You'll find that coverage is paramount — in both clothing and housing.
I live in New Mexico, when out hiking with my dogs I wear a light long sleeved shirt, long pants (I have already survived one rattlesnake bite, I am not up for another), and a wide brimmed hat.
If you are talking about historical movies, it was also cooler back then. The high for June 1895 was 82f/28c July it was 84f/29c, and August was 83f/28c. And in New Mexico our nights are typically 30f lower than our days, and a cowboy back then would have one set of clothes.
Also as mentioned, the sun here is brutal. You have to keep covered or risk sunburn. We also have mesquite with its thorns.
Not a historian but I did grow up on a horse in hot af weather. Long sleeves even in heat for a few reasons: 1 bugs, until you’ve sat on a horse you’ll never realize how many bugs they deal with in life, you do not want to get bit by a horsefly trust me 2 farmers tans suck 3 it’s really not that bad up there even in 100f/38ish c weather 4 you get used to it and the horse is doing most of the work 5 even if you’re not on the horse, you can always roll up the sleeve or unbutton the shirt it’s not like you’re trapped in a long sleeve shirt like it’s welded onto you or anything 6 it only sucks bad when you have to dig fence posts in 100f/38ish c weather, then yeah gotta be careful but generally you did the hard work in morning then kinda slacked off during midday
I grew up in North Dakota though so dunno my knowledge is more like dances with wolves. Our hottest day growing up was 114f/45ish c, I didn’t do jack that day except sit in front of a fan in the house with ac on full blast drinking ice water.
I do miss my horse from back then though, she was the best cow sense horse for wrangling cow herds. Miss ya Smokey.
It's like what Saudis wear in the desert, loose but protective.
a long sleeved loosely woven linen may shield you from the sun, while letting the air pass through
Obviously in movies clothing looks better than it likely did in daily life, which during the “old west” time period would be typically dirty, highlighted with more dirt.
Folks were lucky who had two sets of clothes- most folks had one set and that would include most hired cow hands. Some top men may have two. Three would mean you are very well off.
Sometimes having clothes is better than not in dry heat, as it will protect your skin from the direct sun, and sweat will still evaporate to cool you.
Having a rag or bandana to breathe through when dust/sand is kicked up by livestock translates a bit into the western wear theme as well.
Humidity is different, but the West is usually dry.
Plus it wasn’t quite as hot as it is now.
Men died more back then from everything, so it’s reasonable to believe that many did die from heatstroke, especially in areas where water is in short supply.
But it’s also true that humans are highly adaptable and most of these men were working and living outside, day after day, week after week. They had never felt air conditioning so were likely not in as much discomfort as a soft American would be.
These men often did not lead long lives, and often only did ranch hand work until they could make enough to start their own homestead or business or family. Some were criminals, some just started out poor - think of it as the factory job of that time period. Anyone could learn it- a few were good at it and moved up or started their own ranches.