I'm trying to work out an example of a proper family of elliptic curves in order to get a better grip on the subject. By a proper family of elliptic curves, I mean a proper flat morphism $X\to S$ such that each fiber is an elliptic curve. However, the example staring me in the face seems not to be salvageable. I would like to understand what exactly is standing in the way of this example working out, as well as to be pointed in the direction of some hands on examples of (nontrivial) families of elliptic curves.
The sort of example I feel immediately drawn to is the family $X_t:y^2=x^3+t$ parameterized by $t$ (I understand that $X_0$ is not an elliptic curve, but we should be able to simply remove that fiber). To be explicit with notation, let $X=\operatorname{Spec} k[x,y,t]/(y^2-x^3-t)$ sitting over $S=\operatorname{Spec} k[t]$. The fibers here are not elliptic curves (they're affine), which I would like to solve by moving to the closure $\overline X$ of $X$ in $\mathbb P^3$. Now we are faced with the problem that any morphism $\overline X\to S$ must be constant, as $\overline X$ is projective and $S$ is affine, so we need to extend to a family $\overline X\to \mathbb P^1$. In order for this morphism to be proper, it must be surjective, but it is far from apparent how to map any points to $\infty$ and still have the desired family of curves. It is here that I am unsure of how to proceed, although this family really feels like it should work out. What is going on here?
One of the standard ways of doing this is to start with two general smooth cubics in $\mathbb{P}^2$, given by say $F=0, G=0$. Then we can take the pencil $\lambda F+\mu G=0$, a family of cubics. This gives a rational map $\mathbb{P}^2\to \mathbb{P}^1$ which becomes a morphism after blowing up the nine points of intersections of $F=0=G$ to get a smooth surface $X$. Thus we get a projective morphism $X\to\mathbb{P}^1$ and the general fiber is an elliptic curve.