Finite renormalization freedom in locally covariant quantum field theories on curved spacetime is known to be tightly constrained, under certain standard hypotheses, to the same terms as in flat spacetime up to finitely many curvature dependent terms. These hypotheses include, in particular, locality, covariance, scaling and continuous and analytic dependence on the metric and coupling parameters. The analytic dependence hypothesis is somewhat unnatural, because it requires that locally covariant observables (which are simultaneously defined on all spacetimes) depend continuously on an arbitrary metric, with the dependence strengthened to analytic on analytic metrics. Moreover the fact that analytic metrics are globally rigid makes the implementation of this requirement at the level of local ${}^*$-algebras of observables rather technically cumbersome. We show that the conditions of locality, covariance and scaling, in conjunction with the microlocal spectral condition, are actually sufficient to constrain the allowed finite renormalizations equally strongly, making both the continuity and the somewhat unnatural analyticity hypotheses unnecessary. The key step in the proof uses the non-linear Peetre theorem on the characterization of differential operators.