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.
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
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.