For the abilities, the important need is to disconnect Strength Athletics "mobility" checks (including parkour, climbing, balancing, jumping, landing, tumbling, running, etcetera) from both Strength Weightlifting and Dexterity Sleight manual dexterity.
This mobile Athletics skill requires some physical Strength Weightlifting, such as when hanging by ones fingers, in order to pull ones own weight up over a ledge. Training in Athletics can assume that one is strong enough to maneuver ones own body weight around. But it wouldnt help for Carrying heavy loads beyond ones own body. To be a powerlifter requires training in the Weightlifting skill.
The Dexterity ability continues to represent both "hand-eye" coordination and cautious precision. For example, wielding a rapier is rapid hand-eye coordination. The hand-eye allows for the use of finesse weapons. The rapierfighter can stand with body motionless, while the hand flings the rapier point around. The hand-eye fine motor skills likewise allow for a AC bonus, from blocking attacks (deflecting, parrying, shielding, etcetera), even if the body remains in place.