For any fixed nonzero complex numbers $z_1,\dotsc,z_m$, there are infinitely many $n$'s such that the arguments of $z_1^n,\dotsc,z_m^n$ all lie in $[-\pi/4,\pi/4]$. This follows from Dirichlet's theorem on simultaneous diophantine approximation. For such $n$'s,
$$|z_1^n+\dotsb+z_m^n|\geq\Re(z_1^n+\dotsb+z_m^n)\geq\frac{|z_1|^n+\dotsb+|z_m|^n}{\sqrt{2}}.$$
In particular, the left hand side cannot be asymptotically $1/n!$, because the right hand side is exponentially small at worst.

In short, there is no $\alpha$ satisfying the requirements.