Climbing stairs only helps in endurance. How much weight does your leg carries while climbing the staircase? Only your bodyweight. For squat, the amount can go up to 3 times the bodyweight.

Apparently TS's target is for strength for legs.

Upper body muscle are much smaller than the legs and glutes muscle. It also the reason why most people have lesser bench press numbers compared to their deadlift and squat.

And do check what kind of exercises that those calisthenics practitioner done to achieve their body. Muscle up is the MOST basic exercise for them and what can you do by climbing stairs? One legged hop?