Is there any physically sound argument about why we are allowed to interpret $\dot{\vec p}$ as another force in D'Alembert's principle? In Analytical Mechanics, when we derive the D'Alembert's principle for dynamical systems, we generally argue as;
Since $\vec F^{ext} = \dot{\vec p}$ by Newton's second law, we can interpret it as if 
$$\vec F^{ext} - \dot{\vec p} = 0$$
and see the LHS as a difference of two forces.
Since this analysis leads to a accurate description of reality, I cannot say anything about its validity, but the thing is, in Newtonian mechanics a force leads to a change in the momentum and not the vice versa, so is there any physically sound argument about why we are allowed to interpret $\dot{\vec p}$ as another force ?
Because, all the rest of the analysis about D'Alembert's principle rests on the fact that $W = \int \vec F^{ext} \cdot \delta \vec r$, and $$\vec F^{ext} \not = \vec F^{ext} - \dot{\vec p},$$
so is it just that we are lucky to find the "correct" expression by not being thorough?
 A: Another name for d’Alembert force is fictitious force, or inertial force. These forces arise when the object you are studying is in a non-inertial reference frame. Essentially, when the object is accelerating.
One of the many consequences of being in a non-inertial reference frame, is that performing a naïve work integral considering only ‘real forces’ would not yield the correct answer. This is because, from the point of view of the object you are studying, it is actually experiencing these so-called fictitious forces. If you are in a car in a roundabout, you actually feel as if you are being pushed to the outside bend by a force, which you can’t distinguish from being real. The only reason why these forces are called ‘fictitious’, is because they are forces that arise from changing reference frames, rather than from physical interactions resulting in forces. 
With that explained, what you are doing when you add inertial forces into your calculation, is you are effectively switching reference frames into that of the object in question. By adding the inertial forces, you are essentially making the reference frame pseudo-inertial, for lack of a better term, meaning all your usual equations (such as the work integral) apply. It can also be seen as finding the equivalent inertial reference frame and force combination that describes the same state, and then performing calculations in that equivalent reference frame. 
At the end of the day, it is a matter of how you wish to interpret the maths. This is why you still have arguments today about whether the centripetal or the centrifugal force exists. In reality, it depends on the reference frame you are in, and to observe this you have to consider the inertial forces that arise in these non-inertial reference frames. 
As an addendum, according to General Relativity, gravity itself is a fictitious force, arising from the curvature of spacetime. In general, it is impossible to tell if a force you are experiencing is real or inertial, because you experience both as a force, subjectively. The problems only begin when you switch reference frames, so it all depends on your definitions. 
