Yes, the problem is because you cannot just put a 4-d dynamical spacetime into 3-d spatial metric, one has to put it into the 3-metric and its canonical momentum. However the canonical momentum depends on the extrinsic curvature.
My problem is that in a worldview in which only an instantaneous spatial slice exists, the combination of variables that forms the formula for the extrinsic curvature makes no physical sense. Of course it still does what it does:
but in a worldview of a 3-spatial dimension only, this formula is completely unmotivated and comes out of someone’s hat. Further, in a worldview in which only an instantaneous spatial slice exist, there is not even a notion of “a geodesic of [the 4-d metric of spacetime].”
Au contraire, in a view where the 4-dimensional spacetime exists, this formula appears naturally as a consequence of embedding the 3-spatial slice in the 4-spacetime. This forms evidence in the Bayesian sense that 4-dimensional spacetime exists - similar to how there is Bayesian evidence for theories like inflation and string theory. This is what I meant by “natural”. I hope this clarifies my concern.
You are the expert on A-theory here! I only know that a worldview in which only an instantaneous spacetime slice exists does not jive well with relativity. Of course, as you pointed out, there are A-theorists who do not believe this.