@lemiller thanks for clarifying. however, i'm not sure what you mean by an "asymmetry" in the setup.
in any case, as a first remark, note that _any_ functor $Poly_k \to Alg_k$ admits a left kan extension to a functor $Alg_k \to Alg_k$, just because $Alg_k$ admits all (small) colimits (and $Poly_k$ is small and $Alg_k$ is locally small).
regarding your main question, i think you should first be more precise about the meaning of "$RF_A$". classically, people arrived at derived functors from what might today be considered an ad hoc point of view. for instance, if you have a functor out of a mo…