I wonder if the best way of keeping the d20 dice for resolution, but making stats matter, is to make them straight rolls, adding the relevent stat. Make the DCs something as follows:

DC 10: Easy. Nearly impossible to fail.
DC 20: Moderate. Reasonable chance of failure, depending on the relevent stat.
DC 30: Hard. Requires a good score, or a better roll.

Opposed checks can be a simple roll off (+ relevent stat). This means that the 8 Strength Wizard has an 11.25% chance of defeating an 18 strength barbarian (assuming ties go to the Barbarian) against the 26.25% chance if using a -1 penalty for the wizard on the roll, and a +4 bonus for the barbarian, which (while still not insignificant chance of success for the Wizard) seems better to me.