Why is it impossible to estimate the inclination of an exoplanet with the Radial Velocity Method?
Essentially the problem comes down to geometry: radial velocity provides a 1-dimensional view of a 3-dimensional system. See the diagram below:



The observer sees only the component of the velocity vectors along the line from the observer to the star. That means that they cannot distinguish between the two blue velocity vectors: they will see only the purple (radial) component. This means a larger velocity vector that is tilted further away from the line-of-sight can masquerade as a smaller vector in the line of sight.

No matter how you define your variables, you will always run into this issue because it is inherent to the geometry of the situation.

This does break down in the case of planet-planet interactions, which can break the degeneracy here (a good example of this is Gliese 876).