Earlier quoted context omitted.
I like this way of looking at the problem, I wouldn't have thought of this! I think this relies implicitly, too, on incompressibility of the fluid, no? As well as surface tension? Otherwise, we might imagine the fluid having less density at the bottom, while occupying a circular cross-section of the same area.
It does rely on incompressibility, but, to this order, does not include surface tension. The water faucet puzzle that interested me as a student was after the water had reached the sink and begun to spread out radially from the stream. Have you noticed that there is a circular boundary where the layer of water suddenly changes depth?
Separately, when you model surface tension, you get an equation for when a stream breaks up into droplets - https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_insta...