No: your argument would apply equally well to any family of distributions, not just the family of distributions with finite mean & variance, & it's easy to come up with counterexamples where the sample variance is not sufficient, e.g. the family of exponential distributions with various rate parameters. Sufficient statistics of fixed dimension are updateable (see https://stats.stackexchange.com/q/122917/17230 for why the sample median can never be sufficient) but the converse doesn't follow.

With i.i.d. samples from broad non-parametric families such as the one you specify, the order statistic is minimal sufficient.