Is the amount of blood that flows through every vein and artery per minute really a constant?
I think the question is not whether the blood flux is constant in time, but rather in space: if 5L/min is the flux through aorta (in given physiological conditions), then at the same moment 5L/min will cross the capillaries. This is volume conservation, for a closed system filled with an incompressible fluid, see this sketch:



If 5L are pushed through the main pipe, the same volume needs to be pushed through the system downstream of it.

Then comes the fact that the system is not perfeclty closed, and some water (and solutes) will "leak" in and out, as @inf3rno points, so this is only approximatively true. Another limitation is that vessel walls are actually elastic, so a change in vessel radius does occur during blood flow. However I believe it is a fair approximation.