how is this method of proving shift invariance correct
The so called example system is nonlinear and recursive and those books generally exclude the treatment of nonlinear &amp; recursive systems. Hence you may not find example solutions.

The method relies on the fact that the relationship between the output $y[n]$ and input $x[n]$ remains the same between $y[n-d]$ and $x[n-d]$.

As an absurd example, think of the following. Consider the equation of a circle at origin: $x^2 + y^2 = r^2$. Now, shift this circle to the point $(a,b)$ and re-write its equation: $(x-a)^2 + (y-b)^2 = r^2$. Finally, redefine the variables $x' = x-a$ and $y' = y-b$ and rewrite the equation: $(x')^2 + (y')^2 = r^2$. 

It's seen that the circle is shift invariant (not in the sense of the signal processing context though)