Write $a,b,c$ instead of $\alpha,\beta,\gamma$. Let $p(t)$ be the polynomial in question. Let $$t_*:=\frac{1}{30} \left(15-\sqrt{15} \sqrt{2 \cos (2 a)+2 \cos (2 b)+2 \cos (2 c)+9}\right).$$ Then (under the given conditions on $a,b,c$) $$p(t_*)\ge0. \tag{1}\label{1}$$ Also, $$p(1/5)\ge0\text{ or }t_*\le1/5. \tag{2}\label{2}$$ So, either $p(1/5)\ge0$ or $t_*\le1/5\ \&\ p(t_*)\ge0$, so that $\max_{t\le1/5} p(t)\ge0$. Also, $p(-\infty+)=-\infty<0$.
Thus, $p(t)=0$ for some real $t\le1/5$. $\quad\Box$
Claims \eqref{1} and \eqref{2} can be verified algorithmically, since $t_*$, $p(t_*)$, and the conditions on $a,b,c$ are algebraic in $\cos a,\cos b,\cos c$.
This has been done with Mathematica, which took about 2.5 sec to check \eqref{1} and about 0.5 sec to check \eqref{2}. Here is an image of the Mathematica notebook:
