Why Aperture priority, why not manual ?

We have different modes on the camera which play different roles. M A S P (Nikon) , or M Av Tv P (Canon) symbols are written on the dial on the top of the camera, SLR , compact or Bridge camera. 

Lets understand each on of them.

on the Manual mode, we set both aperture and shutter speed ourselves. We physically change the aperture and shutter peed using two different dials placed on two different places on the camera body.

The Program mode handles everything on its own, so does the Auto mode as well. Do not confuse with A of Aperture priority mode, thats different from the Auto mode.


Manual mode Aperture priority
You need to set both aperture and shutter speed. Only aperture needs to be set, the shutter speed is automatically and precisely set by the camera.
Two different dials located at two different locations need to be rotated, a slower process only one dial needs to be rotated. Faster reflexes help in shooting faster.
what you set physically may not precisely be the same the exact moment you shoot. The exposure is precise.
You set aperture and shutter speed of your choice but you still bring the exposure needle/dot to the middle where it is not over or under exposed, effectively you do the same thing as aperture priority but do it physically. so if you still have to balance the exposure as your first priority, you are not really choosing both aperture and shutter speed of your choice. Most of the times, unless we are shooting a fast moving subject, we look at things in terms of the depth of field. Thus aperture priority is what we have in mind. To get the right exposure, when the camera can set the correct and precise shutter speed by itself, what is the need of physically rotating another dial to set the speed and that too as per what the camera is suggesting. we are still following the camera on the manual mode. 
More chances of exposure variation More precise exposure in most of the situations.
we can make the photograph lighter or darker. with exposure compensation button or dial, we can make the image look brighter or darker, starting off from the exposure recommended by the camera's sophisticated metering.

Why no Shutter Priority mode?

There are more variables of shutter speed than there are for the aperture. so when we set the shutter speed, the camera gets lesser corresponding apertures to give a correct and balanced exposure. so, if the camera needs to open up to f 1.4 but the maximum aperture on the lens is only f 5.6, the resulting photograph will be highly underexposed.    

As the shutter speed variable are more, whatever aperture is set, there is a corresponding shutter speed available. just that at certain aperture setting under lower light conditions the shutter speed may drop lower than the 1/60th sec and a blur may be caused due to the camera being handheld. Just open up the aperture, increase the ISO, or use a tripod in such situations.