If you want other types of skill variance, don't link ability score to skill. Use the Variant rule that says you make Ability Checks and if the skill you have could be applied to it (whatever check it was) then you add your proficiency bonus.
As you said, Athletics could be with STR, DEX or CON. So when the player says what they are doing, you decide whether that would be a STR, DEX, or CON check. And then you or the player makes the call as to whether being a skilled athlete would make the check easier and if so, add the proficiency bonus.
If you are climbing a cliff face? STR check. Add proficiency bonus if proficient in Athletics.
If you are running cross-country for several miles? CON check. Add proficiency bonus if proficient in Athletics.
If you are swinging on a rope across a chasm? DEX check. Add proficiency bonus if proficient in Athletics.
And this can apply to all sorts of things.
Finding your through a huge cathedral? WIS + Religion
Recalling who the god is represented in the stained glass window? INT + Religion
Commiserating with local priest in hopes of getting a bit of gossip? CHA + Religion
If you go about things this way, you not only get all kinds of different types of "total scores"... but you also find several skill end up being unnecessary (I don't use Acrobatics in my games, I use DEX + Athletics.)
As you said, Athletics could be with STR, DEX or CON. So when the player says what they are doing, you decide whether that would be a STR, DEX, or CON check. And then you or the player makes the call as to whether being a skilled athlete would make the check easier and if so, add the proficiency bonus.
If you are climbing a cliff face? STR check. Add proficiency bonus if proficient in Athletics.
If you are running cross-country for several miles? CON check. Add proficiency bonus if proficient in Athletics.
If you are swinging on a rope across a chasm? DEX check. Add proficiency bonus if proficient in Athletics.
And this can apply to all sorts of things.
Finding your through a huge cathedral? WIS + Religion
Recalling who the god is represented in the stained glass window? INT + Religion
Commiserating with local priest in hopes of getting a bit of gossip? CHA + Religion
If you go about things this way, you not only get all kinds of different types of "total scores"... but you also find several skill end up being unnecessary (I don't use Acrobatics in my games, I use DEX + Athletics.)