I think Nintendo has two major problems when it comes to the controllers, and I don't think it has to do with how iconic they are. I agree that it's arguable that the general layout is not quite as iconic as the Dualshock (regardless of its heavy modifications in its newest iteration), with its iconic buttons. That said, I think the NES controller, the SNES controller, and the Wiimote are all effectively iconic enough that their prevalence overshadowed significant changes in design and functionality -- especially with the last one.
Nintendo's first real controller problem is their reluctance to accept the ubiquity of the right analog stick. The Gamecube... sort of, kind of had one, but it didn't really function that well as a right analog stick, and the C-stick was really more of its own thing. The Wiimote deliberately avoided the right analog stick. And then, while the gamepad does feature a right analog stick, it also prominently features motion controls as a method of aiming. While I am sure that there are many that would argue that these methods are superior to the right analog stick, the reality is that the right analog stick is essential to mainstream 3-D gaming.
The second, and much bigger problem, is Nintendo's proclivity towards rushing their controllers. The NES and the SNES are iconic because they were functional and durable. Then came the N64 controller, which was, to use a technical term, a steaming pile of shit. The design was incomprehensible, the analog stick was flimsy, and, of course, it lacked a right analog stick leading to years where 3-D games were awful because of camera controls. The Gamecube controller came next and it was functional and idiosyncratic and I have nothing bad to say about it really. Then came the original Wiimote. I am still, to this day, baffled how popular that system managed to be at launch when it had a controller that did not function. Yes, the Wii Motion Plus fixed a lot of the problems, but that came later. And then there was the Wii U gamepad, which is functional, but it is overly large, cumbersome, has terrible battery life and has a freaking resistive touch screen! Seriously, if Nintendo does not switch to a capacitive touch screeen for the next generation, my fury will know no bounds.