Why didn't he consider the centripetal force as a force acting on that object.
Because Centripetal force is not a force acting on the body. The Centripetal force is not necessarily a unique force in its own right. As an answer pointed out, The "centripetal force" is simply defined as the net force acting towards the centre.
If there were some more forces acting in the radial direction (for example, if your car was tied with a rope to a tree at the centre of the circle, there would be an additional force of Tension towards the centre) then the sum of the forces would be equal to Tension + the Normal reaction component. This sum would then be considered the centripetal force.
Here is another way to thing of the scenario : Write the <PERSON>'s law equation for the radial direction without thinking anything about being on a circular trajectory. Don't think anything about centripetal forces for a moment. 766
If were on any other ordinary track, the only force on the car would be $N\sin(\phi)$ which then by <PERSON>'s law is equal to $ma$. $$Nsin(\phi) = ma$$
Now, because you are moving in a circular path, the "a" in the equation must be towards the centre of the circle and its magnitude is a function of the linear velocity at which you are moving in the circular path (specifically it is $\frac{|v(t)|^2}{r}$ but in your case, v is constant). Now, this is something you have to accept, in circular motion, there must be an acceleration pointing to the centre with the magnitude mentioned above, otherwise the motion would not be circular anymore. Plug this into the equation and you simply get the result as : $$N\sin(\phi) = \frac{mv^2}{r}$$ This pretty much what the other answers said, that the sum of all forces towards the centre (there is only 1 in your case) is the centripetal force. I have taken the opposite sign convention but it doesn't affect the final result.
To answer one of your side questions, if $\phi = 0$, essentially when the road is unbanked and frictionless (as per the diagram), you would never be able to make a turn (In reality, there is always friction to provide the necessary force towards the centre).
Hope this helped. 544