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.

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 the "orthogonal complement" of a (generic) factor $\mathbb{P}^{a_{i_1}} \times \mathbb{P}^{a_{i_2}} \times \{p\}$, then some fibre is smooth hypersurface of multidegree $(1,1)$ in $\mathbb{P}^{a_{i_1}} \times \mathbb{P}^{a_{i_2}} \times \{p\}$ hence not toric by the above. 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, recalling that the fibers of a toric morphism are toric. 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.