Which Flux custom resource applies the Kubernetes manifests found at a path in a source?

Answer: Kustomization (kustomize.toolkit.fluxcd.io)

Explanation

A Flux Kustomization points at a source and a path inside it. kustomize-controller builds the manifests with Kustomize, generating a kustomization.yaml if the directory has none, and applies the result. It is a different thing from the kustomization.yaml file Kustomize reads, despite the shared name.