Why do fluids not accelerate?

The fluid is accelerating. The continuity equation simply states that at any instant $A_1v_1=A_2v_2$ where $A_1$ and $A_2$ are the cross-sectional areas of the upper pipe and lower pipe respectively, with $v_1$ $v_2$ being the fluid velocities in the same. The potential energy of the fluid stored in the upper pipe is being converted to kinetic energy of the fluid flow in the bottom pipe. When the fluid in the upper pipe is at it's highest point, $v_2$ will be the greatest, and gradually this velocity decreases as the fluid height in the upper pipe decreases.

The thing missing from your picture is the steep pressure gradients (and associated acceleration) at the inlet of the pipe. Far from the pipe inlet at the bottom of the tank, the pressure will be approximately hydrostatic; near the pipe inlet, it is anything but. The same thing can be observed at the drain of an emptying bathtub. Once in the pipe, if the flow in the pipe is inviscid and the pipe is horizontal, then it will be at atmospheric pressure everywhere in the pipe (ignoring some subtleties like the finite pipe height in the vertical direction) after the flow becomes parallel, and the flow will not accelerate further.

If there is wall friction in the pipe (which there will be in real pipes), then there will be a force resisting the flow, the result of which is a pressure gradient along the length of the pipe (a force is required to overcome the wall friction). The qualitative picture isn't changed by wall friction however.