Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes

We use a countable-support product of invariant Jensen's forcing notions to define a model of $\mathbf{ZFC}$ set theory in which the uniformization principle fails for some planar $\varPi_2^1$ set all of whose vertical cross-sections are countable sets and, more specifically, Vitali classes. We also define a submodel of that model, in which there exists a countable $\varPi_2^1$ sequence of Vitali classes $P_n$ whose union $\bigcup_nP_n$ is not a countable set. Of course, the axiom of choice fails in this submodel.