Add a seventh ability. Call it Power and say that it starts at 20.
Players use this to roll to hit with their powers only. So damage bonuses are as normal.
Basic attacks would still go off of STR and DEX.
Also if they want to raise this they need to raise it with a stat boost.
It lets you hit "more often" ...Well, all the ideas about lowering monster defenses or giving everyone a static bonus kind of miss the point. No matter what the defenses are or what other static bonuses you get, a higher primary attack stat lets you hit often.
Like some other people have pointed out, I think this house rule would actually cause the opposite problem from the one it seeks to fix. With this rule in place, it's pretty foolish to have a key score that's anything higher than 14

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.