They have a say insofar as they can participate in the same standards process as everyone else. But no, the PSF has no unique say in how PyPI is run, or how pip behaves. This is a pretty fundamental aspect of how Python-qua-ecosystem works.
They have a say in that if it doesn't behave like they want they can point the documentation to something that does. If pip is the tool linked in the documentation it's the official one that has the PSF's blessing, clearly.
We're going in circles. PSF can't unilaterally change any documentation of particular relevance here; the most immediately relevant docs would be controlled by PyPA and PyPI itself. The former has a standards/community review process, and the latter is particular to PyPI.
But again: this has nothing to do with blessings or not. The fact that pip is the official installer and PyPI is the official index does not mean that everything about them stems from an official edict. That's not how Python's community is structured, and it's certainly not how the technical development on anything in Python packaging has ever progressed.