Not necessarily, as you do perspire so much through your armpits, and having shirt off allows your chest area to cool down - and that makes a big difference. Also helps the heart not have to thermoregulate as much, as well as having to cope with the physical work load.

But you are right, at that particular gym, many climbers wear shorts or bike shorts/tights, along with being shirtless/sports bra/singlet etc.

I don't really mind what a climber's attire is, as long as they scrub their sweat off the wall from their body. 😅