Why does an alternate resolution system have to be more complicated than it needs to be? If multiple dice that aren't d20s are inherently more complex they need to ditch the current damage system.
The multiple dice that can be used for damage is more complex than, strictly speaking, it
needs to be. You could, for instance, decide that all weapons and attacks deal d6.
Damage rolls are different than action resolutions, though. For one, different damage dice makes some amount of sense. It seems reasonable that a hit from a greataxe hurts more than a slash from a dirk. Another difference is that damage rolls are fairly self-contained. Players tend to attack with one type of weapon, so they roll the same damage dice each time. Monster stat blocks have damage dice listed right next to the attack bonus, so you don't have to look up any rules that you wouldn't have anyway.It doesn't slow down the game or lead to much confusion. It may be more complex than available alternatives, but I consider it complexity well spent.
Resolving action resolution with a bunch of different dice conventions is also more complex than available alternatives. But figuring out which dice to use slows down play if you need to look it up in the rulebook. It's not something that each player can remember for himself, like damage rolls, because each character is likely to participate in a variety of tasks, each requiring its own dice, over the course of play. You can transition between different tasks, and thus different dice rolls, rapidly and without warning during a game. For people who haven't memorized the whole complex system, that could mean flipping back and forth between relevant pages as actions resolve and players react in unexpected ways. Then you find cases that the rules don't quite cover, and you have to adjudicate which of the many standards you should adapt to the problem at hand.
Much simpler to be able to fall back on "roll a d20, add some modifiers, beat a DC". It also lets you set standard numbers for DCs of a certain difficulty. That simplifies a mess of competing resolution systems for a single table that says "DC10=easy, DC15=intermediate, DC20=complex". That's something you can't do with a system splut among a bunch of different dice systems.