Big Bang Theory: Does general relativity accurately model space's birth-growth?

Clarifying the question

Is General Relativity’s Standard hot “Big Bang” model true, where space came into being from a nothing-point called a singularity? Does General Relativity get the early history of our Universe right?

Structurally, our Universe is same in all directions and in all locations (i.e. homogenous and isotropic). That means a straightforward derivation from Einstein’s gravity equations to our spacetime will inevitably yield the “Big Bang” model of our Universe. Most broadly,

“The Big Bang model describes a universe that is dynamic and evolving, one that started from an extremely hot and dense state at a finite time in the past…” [Ralph A. Alpher & Robert Herman, Genesis of the Big Bang (Oxford, 2001), 29.]

However,

“If we extrapolate this prediction to its extreme, we reach a point when all distances in the universe have shrunk to zero. An initial cosmological singularity therefore forms a past temporal extremity to the universe. We cannot continue physical reasoning, or even the concept of spacetime, through such an extremity. For this reason most cosmologists think of the initial singularity as the beginning of the universe. On this view the big bang represents the creation event; the creation not only of all the matter and energy in the universe, but also of spacetime itself.” [Paul Davies, “Spacetime singularities in cosmology” in The Study of Time III, 78–9., ed. Fraser (Springer, 1978), 78-9.]

This is called the Friedmann–Lemaître–Robertson–Walker (FLRW) Big Bang model:¹

The FLRW Big Bang Model = def. The model, calculated from General Relativity as applied to any homogenous and isotropic universe (like ours), wherein inevitably all space-time reality began at a singularity which ultimately expanded into the Universe we see today. Free floating particles in space with only gravity acting on them would all have a history of traceably moving backwards in time, and ultimately collapsing with space and time itself towards a universal singularity point.² (On this model, no space-time existed outside/before the singularity, so it is meaningless to ask where it happened or what was temporally “before.”)³

Is a model like this, which contains an initial cosmic singularity, true?

1. • Stanford Encyclopedia of Philosophy: “If we push backwards far enough, we find that the universe reaches a state of compression where the density and gravitational force are infinite. This unique singularity constitutes the beginning of the universe—of matter, energy, space, time, and all physical laws. It is not that the universe arose out of some prior state, for there was no prior state. Since time too comes to be, one cannot ask what happened before the initial event. Neither should one think that the universe expanded from some state of infinite density into space; space too came to be in that event. Since the Big Bang initiates the very laws of physics, one cannot expect any scientific or physical explanation of this singularity.” [Bruce Reichenbach, “Cosmological Arguments,” in The Stanford Encyclopedia of Philosophy (Nov 2016)]
2. More technically, all past-directed geodesics terminate a the singularity.
   • Gott et al.: “The universe began from a state of infinite density about one Hubble time ago. Space and time were created in that event and so was all the matter in the universe. It is not meaningful to ask what happened before the big bang; it is somewhat like asking what is north of the North Pole. Similarly, it is not sensible to ask where the big bang took place. The point-universe was not an object isolated in space; it was the entire universe, and so the only answer can be that the big bang happened everywhere.” [1976, p. 65] [3] It is not meaningful to ask what happened before the Big Bang or “where” it took place.
3. Gott, Gunn, Schramm, & Tinsley: “Space and time were created in that event and so was all the matter in the universe. It is not meaningful to ask what happened before the big bang; it is somewhat like asking what is north of the North Pole. Similarly, it is not sensible to ask where the big bang took place. The point-universe was not an object isolated in space; it was the entire universe, and so the only answer can be that the big bang happened everywhere.” [Gott, J. R. III, Gunn, J. E., Schramm, D. N., and Tinsley, B. M. “Will the universe expand forever?” Scientific American, (March 1976). 65.]