Let $X \subset \prod_{i=1}^{n} \mathbb{P}^{a_i}$ be a smooth hypersurface of multidegree $(1,\ldots,1)$. I claim $X$ is toric only if $a_i=1$ for all but one $i$.
For $n=2$, this is Lemma 5 of V. M. Buchstaber and N. Ray, Toric manifolds and complex cobordisms, Uspekhi Mat. Nauk 53 (1998), no. 2(320), 139–140 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 2, 371–373. In fact in this case they are toric $\iff$ $\min\{a_1,a_2\}=1$.
Next, we may apply this to get the general statement. Suppose for a contraction that two $a_i$ are greater than $1$, say $a_{i_1},a_{i_{2}}$. Consider the projection to a (generic) factor $\mathbb{P}^{a_{i_1}} \times \mathbb{P}^{a_{i_2}} \times \{p\}$, such that the image is a smooth hypersurface of multidegree $(1,1)$ hence not toric. To prove the smoothness, apply Bertini to the linear system given by translating the intersections with $X$ with $\mathbb{P}^{a_{i_1}} \times \mathbb{P}^{a_{i_2}} \times \{p\}$ to $\mathbb{P}^{a_{i_1}} \times \mathbb{P}^{a_{i_2}} \times \{p_0\}$ for some fixed $p_0$, if this linear system had fixed part $X$ would contain some product of linear factors, but the restriction to any product of linear factors is a hypersurface with degree $(1,\ldots,1)$. Hence a generic element is smooth.
But this contradictis Proposition 2.7 of https://arxiv.org/pdf/2208.09680. The condition of Proposition 2.7 is equivalent to fibers being connected in this generality, which is true because they are ample divisors in the orthogonal product of projective spaces.