A common-core zero curve exists on a nonempty parameter open

Choose a closed $x\in O$, whose residue field is $k$. First-jet surjectivity gives a $k$-parameter $a$ with value zero at $x$ and derivative $T_{x}O\to E(x)$ of rank two. On a local smooth presentation of $O$ and a frame of $E$, $I_O$ is defined in $O\times T$ by two coefficients. Their derivatives in the $O$ directions are independent at $(x,a)$. Augmenting the smooth presentation's Jacobian by these two rows produces an invertible maximal minor, so the relative Jacobian criterion makes $I_O\to T$ smooth of relative dimension one near $(x,a)$. This argument is valid in all characteristics.

Choose that smooth open neighbourhood in $I_O$. Its image is a nonempty open $T_O$ by [F2]. Every point of $T_O$ has a preimage in the incidence, so its actual zero fiber on $O$ is nonempty, including the generic point of the irreducible parameter space. In particular its closed $k$-parameters have a zero there. This proves the asserted dominance on an actual incidence open and the needed nonempty fibers.