In my view, I use Strength (Athletics) to run across a beam (as if ‘climb’), for the same reason I use Strength (Athletics) to aim a sword or throw something heavy.
I think it depends upon how you see the definition of athletics. For example, some people call acrobats athletes.
Athletics necessarily includes gross motor skills, body coordination, and physical stunts such as jumping.
I disagree. It think it depends upon where you draw your lines.
For me, athletics and strength is to be used when you can muscle a solution. Acrobatics and dexterity are to be used when finesse is required. Many solutions can be addressed by either. However, some problems require one solution or the other. Basically, it is up to the players to explain how the PC uses their skill to solve a problem, so it really comes down to their description.
It seems to me, the early 1e decision to link Dexterity to balance screwed up all of the ability scores ever since. People who are master archers, who stand still and aim carefully, are never necessarily master gymnasts.
No, that would be covered by their proficiency, not their basic ability score.
And each of these skills is a massive simlification. Intelligence (history) is a skill. You roll it for checking your knowledge of any type of history. However, in the real world, there are a lot of people that know a lot about US history, but nothing about Spanish history.
I once tried to add depth to the skill system by giving people 'specializations' in certain uses of skills. For example, you might have bonuses on strength checks using upper body strength, or history checks for local history. It just became too much to manage and we went simpler. That was in an eearlier edition, but WotC followed my approach with 5E - keep it simple.