Wow, I found the writing style to be extremely grating, but here's my attempt at correcting some faulty premises that lead the author astray:>narrator: he’s assuming that the water at different vertical positions, at a given horizontal position, is moving with the same vector
This assumption is physically impossible, given the first diagram in part #2. Obviously in real life the thing would drain out the side, so obviously there isn't a uniform flow gradient, neither top-to-bottom nor radially. The flow has to bend somewhere, after all. This sometimes happens where students get so wrapped up in the math that they lose common sense and "prove" reality wrong, creating a paradox.
The author does it again with the "top half of an hour glass" shape:
>Well, we know that the velocities are always strictly vertical (and downward).
No, we actually don't know this. This is again physically impossible, given the shape of the container. If there was no horizontal component you'd get a tube-shaped void extending up from the spout, and water would magically fail to drain from the sides as if you'd drilled out a core sample from solid ice. Obviously, this is not what happens in reality, so obviously the assumption that all the velocity is purely vertical is incorrect.
>The water at the bottom of the spout cannot be going faster than at the top! [...] In other words, the water does indeed accelerate, and it also gets narrower; the pipe is partly filled with air.
The author appears to have forgotten that friction and gravity exist. Generally if the flow is steady (which it will be unless the volume of water in the tank is small compared to the volume of water in the outlet pipe), then the pipe will run full of water at all times. Water does not accelerate in the pipe. If you add length to the pipe then the added height is counteracted by added friction and by the weight of the moving fluid.
Grab an undergraduate fluids textbook and do an energy balance.