Extra dampening from where ever doesn't matter if we're talking about stopping on a dime and making a perfect square wave. I'm also not talking about driving the sub to or beyond mechanical limits -- we're talking about feeding the sub a clipped signal whose power is within thermal and mechanical limits of the sub.
But yes, the car analogy wasn't the best but not for reasons that you mentioned. The reason why is because all the dampening factors you can think of become arbitrary if you're trying to say the sub will make a perfect square wave. Why is this? Because you have an object traveling at the speed of light. This object is the moving assembly of the speaker. Are you familiar with the theory of Special Relativity and basic Calculus?
Using these ideas, I can prove that it will not move in a perfect square wave and that it would require infinite acceleration. This is NOT an opinion; this is math and physics.
Starting with the calculus approach, if you look at a square wave in a position vs time graph, you can see it goes from one point to another in 0 time.
Just like this image I made, the speaker cone is expected to go from at rest to, let's say, 1" excursion. It is then supposed to stop on a dime and hold that 1" excursion point perfectly for 1/2 the wave length. At that point, it travels from the 1" point to the -1" instantly. This is 2" of travel done in 0 time and we did 1" of travel in 0 time as well. How can you travel different distances when taking the same amount of time? Well your velocities would have to be different. Here though, they don't matter because you get there instantly. To understand instantaneous travel, we have to just to the relativity discussion but I'll wait.
The next graph we need to think about is the velocity vs time graph which would be the derivative of the position vs time graph shown above. The only problem is that we have a function with an infinite slope (rise/run and in this case 1"/0). If the slope is infinite, the derivative is infinite which makes the next derivative (acceleration) infinite as well. That means we need infinite velocity to accomplish this goal. We need to accelerate to get to that point and the acceleration would also be infinite. This all would be possible if the cone could travel at the speed of light though.
This is where relativity comes in. Time dilation is a very real part of special relativity that states as an object approaches the speed of light, time begins to slow down with respect to the object traveling at those speeds. This happens to the point where time actually stops when you reach the speed of light. Well hey, that same thing is happening here! Time apparently stops for speaker since its moving assembly travels at the speed of light to travel a distance in no time. This is what you're implying if you think the sub actually follows the square wave with no overshoot and perfect hold.
So like I have said before, there will be overshoot and there will not be perfect hold. It's NOT an opinion that I have, it is a FACT. The sub will still move up and down and it will not travel less distance or be frozen as if you have a battery on it. So the SIGNAL (which is what we're talking about here) will not reduce the thermal power handling of the sub by a significant amount.
As for your mechanical reasoning, think about a speaker playing more than one frequency at a time. Think about the sharp peaks and quick transitions needed to play them all at the same time and then to change to other signals quickly. Does this ever cause speakers to tear themselves apart? Of course not. Speakers are capable of playing all sorts of signals without tearing themselves apart, including clipped signals. The only way they would tear themselves apart is if they were accelerating at an infinite rate!
