Just to point out something that always bothered me: there are no bodybuilders winning olympic swimming competitions - no swimming medalist packs on muscularity like Arnold S. (aka Conan) - it was always a mistake to attribute the skill of swimming to strength and, considering the intense cardiovascular training involved in swimming, Swimming as skill IMHO should rely on Constitution.
I like Running as a skill relying upon CON too - makes sense to me as it too is cardio intensive.
Edit: are there not trades and crafts that need strength? Smithing, longshoreman, mining? Or is that too niche?