Ok, thank you! How about if we assume that $\alpha_i\neq 0$, $\beta_i\neq 0$, $\gamma_i\neq 0$ for all $i$?

You are absolutely right. Anyway, my main concern is to find (if there exists) a triangular similarity transformation which normalizes $A$. Now, if we restrict the attention to the subclass of normal matrices which are circulant, this is not possible (for every choice of the parameters). But what can be said about the general case? (I edit the question accordingly.)

I think I missed your point. If $p(s)=s(s-1)^2$ then, by applying the Routh-Hurwitz stability criterion, $p_\varepsilon(s)=s(s-1)^2+\varepsilon$ has always a root with positive real part for all $\varepsilon\in\mathbb{R}$.