The VSI (and the altimeter) uses the pitot-static system, but not the pitot tube.
The pitot-static system consists of a static pressure port, the pitot tube, and (on more sophisticated aircraft) a computer for interpreting the readings from the two sensors. The altimeter and VSI only rely on the static port (VSI is just the derivative of altitude with respect to time), while the airspeed indicator relies on the satic port and the pitot tube in combination.
If the static port has a problem, none of those three instruments will work properly. Whenever I've seen this happen, altitude remains stuck at some fixed number (usually sea level, but not always), VSI is zero (becuase the derivative of a constant is zero), and airspeed is way off (but continues to change).
If just the pitot tube has a problem (as was the case in this mishap), the airspeed indicator won't work (usually stuck at some fixed number, most often zero) but the alitimeter and VSI will be fine. The VSI and altimeter were both telling them that they were falling out of the sky like a rock, and they should have paid attention.
If they were unsure as to whether it was a problem with the pitot tube, the static port, or both, they had a tie-breaker: the stall warning system uses a completely different set of sensors (they measure the airflow over the wings, looking for signs consistent with a stall). The stall warning system was telling them they were in a stall, the VSI and altimeter were giving them information consistent with a stall, and the airpseed was not changing as it should have in response to low-throttle and nose-high. Put those together and the most logical explanation is that the airspeed indicator is full of shit. That's what I was talking about when I said that human pilots have (or should, at least) the ability to figure out the truth when multiple sensors are in disagreement.
An analogy: there's an old adage that you shouldn't go sailing with two compasses: three, or one, but not two. The reasoning is that if you have two compasses and one goes bad, you won't know which one to trust, so you might as well have none. If you have three, you can trust the two that agree with each other. A lot of avionics systems operate on this principle: three sensors, with the results combined by a voting system. Computers do this well. However, when you have a human in the loop, with access to information the computers don't have, this adage is actually a bunch of bullshit. If you have two compasses and they disagree, you're better off than if you only had one because at least you know immediately that something is wrong. Then you can pull in other clues to figure out which compass is right: look at the sun, or the stars, or rely on your knowledge of prevailing wind and current conditions in your general area. If the compasses disagree by more than a few degrees, you'll be able to quickly figure out which is wrong. If they only disagree by a few degrees, they'll probably get you safely close enough to land that you can go back to navigating by dead reckoning, and then figure out which compass is wrong.