Even the GPS REQUIRED notation might be ambiguous depending on your aircraft documentation.
For example the generic boeing 737 FCOM tells us the following about GPS approaches:
GPS approach - an approach designed for use by airplanes using stand-alone GPS receivers as the primary means of navigation guidance. However Boeing airplanes using FMS as the primary means of navigation guidance, have been approved by the FAA to fly GPS approaches provided an TNP of 0.3 or smaller is used.
Since that is achievable without GPS updating this excerpt actually allows us to fly GPS approaches without having a GPS, for example in 737 classics without GPS retrofit.
Granted, that is only for approaches and not arrival procedures. however as you correctly say GPS is not required for P-RNAV certification, no clue about RNAV 1 though.