It is even easier than outlined in my comments. Without loss of generality, assume that $k$ is an algebraically closed field. Let $X$ be a $k$-scheme that is an Abelian variety. Let $L$ be an invertible sheaf on $X$ such that the associated $\mathbb{G}_m$-torsor $Y:= \underline{\text{Spec}}_X \bigoplus_{d\in \mathbb{Z}} L^{\otimes d}$ over $X$ has the structure of a group scheme such that the projection $\pi$ to $X$ is a morphism of group schemes with kernel subgroup scheme equal to $\mathbb{G}_m$.
Lemma. For every $k$-point $x$ of $X$, there is an isomorphism of $Y$ as a $\mathbb{G}_m$-torsor over $X$ with the pullback of $Y$ by the translation $\tau_x$.
Proof. Since $\pi$ is geometrically surjective, for every $k$-point $x$ of $X$, there exists a $k$-point $y$ of $Y$ such that $\pi(y)$ equals $x$. Since translation $\tau_y$ by $Y$ and translation $\tau_x$ of $X$ commute with $\pi$, i.e., $\pi\circ \tau_y$ equals $\tau_x\circ \pi$, the translation $\tau_y$ defines an isomorphism of $Y$ with the pullback $\mathbb{G}_m$-torsor $X\times_{\tau_x,X,\pi} Y$. QED
Denote the group operation on $X$ by the following morphism, $$ m:X\times X\to X.$$
Theorem. The pullback $m^*L$ is isomorphic to the tensor product of $\text{pr}_1^*L$ and $\text{pr}_2^*L$.
Proof. By the lemma, for the invertible sheaf $M:= m^*L \otimes_{\mathcal{O}_{X\times X}} \text{pr}_2^*L^\vee$, the adjunction homomorphism from $\text{pr}_2^*(\text{pr}_2)_*M$ to $M$ is an isomorphism from the pullback of an invertible to $M$. Thus, $m^*L$ is isomorphic to the tenso product of $\text{pr}_1^*L$ and $\text{pr}_2^*L’$ for some invertible sheaf $L’$ on $X$. Finally restrict to a fiber of $\text{pr}_1$ to deduce that $L’$ is isomorphic to $L$. QED
Because of this, the induced morphism of Abelian varieties,
$$
\phi_L : X \to \widehat{X}:=\text{Pic}^0_{X/k}, \ \ x\mapsto L^\vee \otimes_{\mathcal{O}_X} \tau_x^*L,
$$
is the zero morphism.
In characteristic zero, you can now finish the argument using the analytic theory of holomorphic line bundles on a complex torus, cf. Claim (iv) on p. 84 of Mumford's "Abelian varieties."
In positive characteristic, it is trickier. The finiteness theorems from SGA 4 imply that the torsion subgroup of the Néron-Severi group $\text{Pic}(X)/\text{Pic}^0(X)$ is a finite torsion group (cf. Kleiman's exposé in SGA 6 for a detailed explanation). If we admit that these invertible sheaves are "algebraically trivial" (by which you must mean that some positive multiple is algebraically equivalent to the trivial invertible sheaf), then the problem now is to prove that every nonzero element of the Néron-Severi group is not numerically trivial. In Section 21 of Chapter IV of Mumford's book, he proves positivity of the Rosati involution. In particular, this implies that the map from the Néron-Severi group to the associated $\mathbb{Q}$-vector space of the endomorphism ring of $X$ has kernel equal to the torsion subgroup of the Néron-Severi group, where the homomophism sends each algebraic equivalence class $[M]$ of an invertible sheaf to $\phi_A^{-1} \circ \phi_M$ (where $A$ is a fixed ample invertible sheaf on $X$). In fact, the $\mathbb{Q}$-vector subspace generated by the image is exactly the fixed locus of the Rosati involution, but this is irrelevant to the proof. What is relevant is that an invertible sheaf $M$ is in the torsion subgroup $\text{Pic}^\tau(X)/\text{Pic}^0(X)$ of the Néron-Severi group if and only if $\phi_M$ is the zero homomorphism. Thus, a positive tensor power of $M$ is algebraically equivalent to the trivial invertible sheaf if and only if $\phi_M$ is the zero homomorphism. By the lemma, it follows that a positive tensor power of $L$ is algebraically equivalent to the trivial invertible sheaf.
