I'm talking about mouse movement as well. I never mentioned a gamepad. And I use time-scaled mouse movement, and I don't have the issue the OP is having.
And changing a line of code resulting in a fix is not actually a proof that the line contained the bug.
No, time shouldn't be involved because regardless of what frame you are on or how fast your FPS is, the change in mouse movement (deltaX/deltaY) remains constant.
Say I move my mouse 2 inches to the right at a constant speed; to make it simple lets say it takes us 1 second to do this.
At 30fps, each frame will calculate deltaX as 2/30 or 0.0667 inches.
At 60fps, each frame will calculate deltaX as 2/60 or 0.0333 inches.
Using 60fps we simply calculate distance at a faster rate than we do at 30fps, but that doesn't change the overall distance I moved or the time it took to get there.
If you imagine a line representing the distance my hand moved, and ticks on the line representing where the program calculated deltaX, the 60fps program will simply have more ticks and they will be spaced closer together. But the distance I traveled and the time it took to get there remain constant.
As a matter of fact I'm not even sure if this would be used in joystick movements either, as you would simply multiply a movement speed times the value of the X axis of the stick ranging from -1.0f to 1.0f. Unless I'm misunderstanding how the hardware is used.