I've done some Diet Dr. Pepper insomnia fueled thinking and I've decided to ditch the Knave-style resolution system I described earlier.
Instead I've opted for a Target 20-style system. The basic formula is d20 + Score ≥ 20.
The reason I like this system instead is because it gives an average level 1 PC about a 50% chance of success. Plus it's easy for a player to grasp. If they have an average score (10) all they need to roll is double digits to succeed.
Another reason I chose this over Knave's is because I really prefer the Arnold method of rolling 4d4 for ability scores and using the divide-3-minus-3 formula for modifiers.
Also a Target 20-style system establishes that the score is for the d20 and the modifier is for everything else.