Strength and agility are not the same, cannot be swapped interchangeably, and feats of strength can't accomplish feats of agility and vice versa. Every example the OP gave is someone who has BOTH.
Basically this. But to elaborate as a part time sportsman myself (swimmer, distance runner, table tennis)......
All the agile people in the OP photos are "strong". Not all the strong people out there are agile. Especially when you compare the power to weight ratios of different sports people. Take the strongmen and the gymnasts are two opposite sides of the equation. Now, if we really want to make a proper ruling here, interconnection is the word we are looking for (as mentioned before). And there is one down to the point example in DnD i always like to mention and i think makes the perfect analogy. The LONG BOW in DnD. How come a weapon that had a draw weight of over 100 pounds can be considered a dexterity weapon? The plain and simple truth is, it can't. You must be one strong dude (or dudette) to ever pull the thing effectively. But you also need good hand-eye coordination and a steady grip to be any good with it. So it takes both. Just like gymnastics.
The real question that bothers me though is the unification of agility and dexterity. While the former requires some minimum level of strength and a some minimum level of body wide muscle control, dexterity in most of its incarnations require almost none of the above (paining, sculpting, aiming a weapon, throwing light objects at short range...). But can assure you having the ability to paint a perfect replica of Mona Lisa, or Dismantling a lock won't help you one bit when doing somersaults. Strong core muscles, arms and legs will. Flexible mid and lower back will. Painting skills??? Not so much.
So, from a rule based context, what we as nerds need to develop is a system that would reflect what we observe in reality. For a long time i was a proponent of making Str, Agl and Dex separate attributes...... But as i wrote this block of text, i got another idea..... What if Agility is a derived value? Like AC is. Let's say you have 2 basic abilities, strength and muscle control (dex). Each of these would give you an arbitrary bonus to your total agility. Agility which you can use to add to your dodge actions. Other actions, like parry or aim, will count on your dex bonus alone.
We could probably go this way for other attributes as well.
EDIT: I would put the manual aspects of dexterity almost completely in the skill tree (disarm traps i.e)