> Ok, but if the water is faster at the bottom of the long spout… We could view the top part of this system as an exact copy of the short-spout version. At the interface between the tank bottom and the pipe-spout, the velocity of the water should be the same as in the no-pipe version, right? But that means the water inside the pipe is accelerating inside the pipe:

No, it's not the exact same. In the top part of the long-spout system there's a lack of airpressure holding the water above it back compared to the short-spout, and quite a bit of cohesion in the water pulling the water above it down faster if the lack of air pressure isn't enough. The water in the whole system moves faster as a result.

You'd theoretically get the air (actually vacuum) bubble if you ran the experiment in a vacuum with a liquid that has no cohesion... liquids with no cohesion are otherwise known as gasses though and behave differently in other ways as well.

What kindergarten did you go to? Maybe my public kindergarten education was seriously lacking.