Yes, it does not work. According to the OP article, Boeing said to American Airlines pilots in November that the warning would work on the ground, but later said it in fact does not.
By before takeoff, I assume that means "during the takeoff roll"
And I hope that also means "before V1".
I know the DC 8 couldn't do it's elevator check until it was moving quite fast, as there was no direct connection between the pilots controls and the elevator and it couldn't respond without airflow.
What speed is that? I don’t know about 737s in particular but some AoA sensors work at speeds as low as 20 knots—far less than V1 on any runway a 73 is going to use.
So if you were to get a value that doesn't make sense on the ground when you aren't moving, that would be a assuring sign that you have correctly detected a fault.
The angle of attack of a wing that has zero airspeed is undefined. You could take two traditional weathervanes and put them inside your house and they wouldn't tell you the direction of the wind outside.
I doubt that, the alpha vanes aren't particularly useful until the plane is actually moving at a fairly decent clip.