Maybe one could also give an argument along the following lines:

Given an elliptic curve $E$ over some base scheme $S$ that satisfies $H^0(S, O_S) = k$. Choose an integer $N$ such that $N \ge 3$ and $N$ is prime to $p = char(k)$. The group scheme $E[N]$ is finite and etale over $S$, so we find an etale covering $S' \to S$ sucht that there exits an isomorphism $\alpha \colon E[N] \times_S S' \simeq (Z/n)^2$. We say $\alpha$ is a level-$N$ structure on $E' = E \times_S S'$. By basic results form Katz and Mazur, there exists a fine moduli space of elliptic curves with level-$N$ structures: $M[\Gamma(N)]$ with a universal family $(E^{univ} \to M[\Gamma(N)], \alpha^{univ})$.

However, $M[\Gamma(N)]$ is affine, so the map $S \to M[\Gamma(N)]$ given by the pair $(E, \alpha)$ factors over the affine hull of $S$, i.e. $Spec(k)$.

It follows $E' \simeq E^{univ} \times_{M[\Gamma(N)]} Spec(k)$, thus we get a product family after etale basechange.

Maybe one could also give an argument along the following lines:

Given an elliptic curve $E$ over some base scheme $S$ (edit: proper reduced and connected) over some field $k$. Choose an integer $N$ such that $N \ge 3$ and $N$ is prime to $p$.. The group scheme $E[N]$ is finite and etale over $S$, so we find an connected finite etale covering $S' \to S$ such that there exits an isomorphism $\alpha \colon E[N] \times_S S' \simeq (Z/n)^2$. We say $\alpha$ is a level-$N$ structure on $E' = E \times_S S'$. By basic results form Katz and Mazur, there exists a fine moduli space of elliptic curves with level-$N$ structures: $M[\Gamma(N)]$ with a universal family $(E^{univ} \to M[\Gamma(N)], \alpha^{univ})$.

However, $M[\Gamma(N)]$ is affine, so the map $S' \to M[\Gamma(N)]$ given by the pair $(E, \alpha)$ factors over the affine hull of $S'$, i.e. $Spec(k')$ where $k'$ is a finite extension of $k$.

It follows $E' \simeq E^{univ} \times_{M[\Gamma(N)]} Spec(k')$, thus we get a product family after etale basechange.