Sorry but you need to be a little bit more careful about the bend-and-break. It only works if you can deform the elliptic curve with a fixed complex structure. But if you can vary the complex structure, then it will specialize to a nodal rational curve. In this case you are still fine.

In case of $g=1$, the condition implies that there is a curve of arithmetic genus 1 passing through 2 general points. If this is a irreducible embedded curve, then you can deform it with 1 point fixed. Then bend and break tells you that there is a rational curve through the fixed point. If this is not an irreducible smooth embedded curve, then there are components of genus 0, passing through 1 of the two general points. In any case, there is a rational curve through a general point. In general, I think you need a genus g curve with g+1 points.