The introduction of the Service-Based Architecture (SBA) for the 5G Core Networks has drastically changed the way these networks are designed and operated. Aiming for higher flexibility and agility, the adoption of SBA is the first step towards cloud-native deployments of 5G Core. However, the high degree of functional decomposition in SBA has implications in terms of increased inter-NF signaling traffic during the execution of control plane procedures, as well as an increased complexity in orchestrating a system with tight inter-NF dependencies. In this work, we introduce PP5GS as a stateless 5G Core architecture that implements a procedure-based functional decomposition of the 5G Core NFs. We develop Per-Procedure NFs for four different control plane procedures and perform extensive evaluations in a private cloud environment orchestrated with Kubernetes. The results show that PP5GS requires up to 34% and 55% less computing resources compared to the baseline stateful and stateless systems, respectively, while generating at least 40% less signaling traffic. Moreover, complex control plane procedures can complete up to 50% faster. Lastly, the results show that PP5GS is a more feasible architecture in leveraging edge-offloading of 5G Core NFs.