General Relativity is a theory of gravitation, not nature! Not quantum mechanics! Not even inertia or the fundamental origin of inertia and gravitation.
And vacuum solutions, like those resulting in Schwarzschild and Kerr black holes, are hypothetical extensions of general relativity, not real world applications of the field equations. They are what ifs, that help us understand some of what might be, under conditions that do not exist.., and where they represent approximations of what may exist, the details of everything inside of their predicted event horizons, are far beyond our abitplity to observe and/or confirm. Gravitational dynamics of observed stellar orbits outside of the predicted (but unobserved event horizon), are well described by the field equations of general relativity...., and do not involve infinities.
It seems there is a great deal of confusion about what is a core part of a theory and what is an extension and/or hypothetical application of the underlying theory. By their very nature all vacuum solutions are what if hypotheticals... They assume conditions that do not exist anywhere in our reality.
GR is not and never was intended to be a TOE! Not even a theory of almost everything... Just a theory of gravitation.