As far as I know there are three basic resolution systems in 2nd ed. AD&D. The first is the simplest, roll over X on a D20 to generate a pass or fail. (Saving Throws), The second is almost identical, roll under X on a D20 (or D%) to generate a pass/fail. (Ability Checks, Proficiency Checks, Thieving Skills) and finally the To-Hit/Damage mechanic, roll of X on a D20, with the degree of success being based on a second die roll with a situationally determined die-type.
There are of course outlier mechanics (Roll X on a d6, like the elves ability to detect secret doors), but the above three are by far the most representative of the system, I think, and the to-hit/damage roll is both the most iconic and the most versatile.
So, that'll be the resolution mechanic right there. Roll 1d20, try to get X or higher. If the roll is higher than X than you reduce the challenge your confronted with by XdX points.
Rolling multiple dice takes more time.
The d20's linear progression and unpredictability.
The d20 result doesn't directly effect the degree of success (the "damage dice")
The use of polyhedral's.
My solution to the above is as follows:
Roll 3d6 (totaled) and add any relevant modifiers. If the result is greater than X (the "AC") than the action succeeds. The, the value of 1, 2 or all 3 dice are totaled to determine the impact or effectiveness of the action, based on the situation (the "damage").
So that's the system that I'm going with.