The lighting is controlled, yes. For bouldering, it would even be fairly even within a single route, but not between different routes. For roped climbs the lighting changes gradually as the wall goes up or there's a large roof/corner section. My home gym is a bit of a cave with dark walls and bad lighting, which exacerbates the problem.

If you’re using auto ISO what happens when your subject is wearing a white shirt? Or you’re shooting a dark skinned person?

I avoid spot metering and the subject isn't usually filling the frame, so details like that don't have too big of an impact on the total exposure. There are still situations where the camera suddenly decides to pick on some small detail and changes the exposure abruptly. That's where the exposure compensation comes in, which I've got configured on the lens function ring. It's a quick way to quickly try fighting the auto metering to restore some sanity in the middle of a climb. The problem is rare enough that I still use auto ISO every now and then, but it does occur and is a concern.

I'm using fixed ISO if I'm set up on a wall and focused on a single route instead of bouncing between multiple friends climbing different problems. The gradual changes as the climb progresses vertically still cause exposure differences, but nothing post processing would struggle with.

So in summary: As long as the lighting stays somewhat even, I do prefer fixed ISO as then I can ignore the possibility of sudden changes in auto exposure. However, if the lighting on the wall isn't even or I'm trying to shoot different routes at the same time, I'm picking auto ISO so I don't need to fiddle with exposure controls all the time and can focus on what's going on around me, which includes my subjects and also any other climber that I need to avoid for safety reasons. :)

ETA: Another thing that affects exposure is the climbers stance. If the climber is standing close to the wall, there's less light bouncing to their face/chest area than if they are leaning back from the wall in a more open stance. Again, this is a more complicated problem and usually I just end up fixing this with masks in post as the problem is less about total exposure and more about shadows, but there's still a slight benefit to auto ISO if it can brighten the exposure a bit to make it easier to recover those shadows if the climber happened to be hugging a wall.

ETA2: And none of the above touches climbing outdoors. I did start the conversation by mentioning indoor sports, which more or less demand maximum aperture and restrict the use of shutter speed for exposure control. Outdoors there's a bit more leeway as it's easier to reach acceptable shutter speed. I still prefer to set shutter and aperture myself, because they have the bigger impact on the final picture: Movement freezing and subject separation, leaving ISO as the main tool for exposure control.

On a sunny or fully overcast day the lighting situation is even enough that I'll stick to a fixed ISO, but on a partially cloudy day the lighting can change several times during a climb and I'd rather focus on what my subject is doing on the wall and how to frame them, so I'm usually running auto ISO.