You can't "double-tap" a stick.

Double-tapping in one direction is a common control input for running/lunging/dodging, and has been for a long time.  At least, I remember it was in Kirby's Adventure for the NES, and I'm pretty sure it was around before that. (Any game historians know for sure?)

Problem is, back when double-tapping came around, we were using D-pads.  Now we're using analog sticks, but we're still double-tapping, thanks to convention, and it drives me up the wall.

So what's wrong with double-tapping with a stick?

It's an input gesture designed for directional pads.  Double-tapping on a D-pad makes sense.  Directions are push buttons, so you spam them a little.  It's what your right hand is already doing.  It feels natural, because buttons are designed for pushing.

The analog stick is designed to communicate direction and magnitude based on its position.  The idea of double-tapping a stick doesn't flow naturally from that usage; it's shoehorned in because it's a convention.

It's slow.  Double-tapping of any sort is already subject to input lag from the time between the two taps, but double-tapping a stick is slower than double-tapping a D-pad, since (a) the thumb has farther to travel, and (b) the thumb is used for all three steps of the motion (forward, neutral, forward).  (When double-tapping a D-pad, the thumb is responsible for the two pushes, but the release is largely handled by the inherent spring mechanism of the button.  The springs in the analog stick do assist the return towards neutral position, but I do notice a difference in effort between that and a simple button release.)

This slowness becomes a big problem in fast-paced games where double-tap is used for a dodge action.  There is a danger that either dodging will be useless because it can't be executed quickly enough to react, or that enemy attacks will be telegraphed to an excessive degree to make time for the dodge reaction, slowing the pace and damping the intensity of the game.

It's exhausting.  Yes, I realize I'm describing the slight motion of a single thumb as "exhausting", but I'm talking in comparison to double-tapping a D-pad.  It's exhausting for the same reasons it's slow (distance and effort), but I also find it's a more unnatural motion for the thumb (rapid side-to-side movement).  Granted, double-tap inputs are usually far less frequent than button presses, but over a long gaming period, I find my double-tapping fatiguing far quicker than my button spamming.

It's unreliable.  While it's not exactly rocket science to detect a double-tap on a stick, it's basically foolproof on a D-pad, since all you have to check for is ON-OFF-ON over a short time frame.  The simplest check on a stick would be to set some threshold value to distinguish ON and OFF, but in practice, no two double-taps are the same.  Depending on the position of the stick before the input, and the position of the thumb on that stick (which can change often throughout play), the input may end up being a little too slow, or miss the threshold, just because the particular circumstance called for a more difficult or unnatural thumb motion than usual.  Make the threshold or timing too lax, and you get false positives.

More complex control logic can do a better job of discerning the player's intent in varied circumstances, but why go through all that effort for an input gesture that's unnatural, slow, and tiring?

It's useless.  This is simply what follows if you have any input gesture that's unnatural, slow, tiring and unreliable, because players will avoid using it.  If you implement a mechanic, but discourage its use, what good is it?  At best, it's a waste of time, and at worst, it breaks the game if that mechanic is crucial to your overall design.

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I know convention is useful in easing a game's learning curve, but I'd much rather see a new convention take the place of double-tapping on analog sticks.  Or even if there's no convention to replace it, and each game does its own thing that's right for that particular game, that's fine too.  I just kind of want to, you know, never have to double-tap a stick again.