The renormalization group (RNG) analysis of the Kosterlitz-Thouless (KT) phase transition is a basic paradigm in statistical physics and a well-known success of the RNG approach. It was recently shown that the derivation of the RNG parameter flow for the KT problem rests on the assumption of one-sided polarization, which is implausible and cannot be derived from first principles. We extend this analysis by exhibiting simple self-consistent alternate assumptions that lead to different parameter flows. We then study the KT transition numerically and show that the properties of the transition are well described by the standard RNG analysis (up to some minor paradoxes). The reason for the success of the assumption of one-sided polarization remains unknown; the problem exemplifies the difficulties in the mathematical analysis of the RNG. Related issues in turbulence theory are pointed out.