There's no real requirement that stats be required for a class, period.
If stats were basically perks instead of requirements, such that you could have viable if sub-par characters with six 10s (making NPCs super easy to work with), you could give your character any stats you want and still be viable.
You would have to:
* Make every stat equally useful to a well-rounded character of any class (specialist characters can still gain by specializing).
* Make every class able to effectively utilize every stat (wizards would need VIABLE melee attacks, etc.)
* Make every bump in a stat useful (if 12 is no better than a 10 nobody will have a 12)
* Make every bump in a stat equal to a bump in another stat
* Remove ability scores from to-hit and defense values. Those need to be baked in elsewhere.
* Level out accuracy. Every class should be able to hit with any ability they can access. A wizard whacking someone with a staff may do relatively paltry damage, but they should be able to hit reliably hit so they can utilize Strength.
Just as a 1:30am no-effort example:
"Bump" defined: The point in an ability score increase where you actually gain something from it.
Strength: +1/bump to melee damage
Dexterity: +1/bump to ranged damage
Constitution: +1/bump to HP
Intelligence: +1/bump to area damage
Wisdom: +1/bump to minimum roll on Skill Checks (Wisdom 12 means your lowest skill check roll is 2, etc)
Charisma: +1/bump to HP healed (Wisdom 12 means you get healed for 11 instead of 10, etc)
Obviously the above is not likely going to work for D&D, and a strict +X*bump structure is not the only way to go, but as a general notion, I hope that gets the idea across. You also don't have to make the stats mean the same thing for ehvery class, but making a unique series for each class can get complicated as heck.