> Mathematically, the universe looks a little bit like a black hole running backwards in time

It really doesn't, not even a little bit.

Schwarzschild has spherical symmetry, Kerr has axisymmetry, neither has homogeneity, and each has isotropy only at a very restrictive set of points in spacetime. These are vacuum solutions of the Einstein Field Equations of General Relativity; comparable metrics that introduce collapsing matter (e.g. Lemaître-Tolman-Bondi in the spherically symmetric case, something like a perturbed LTB or perturbed Oppenheimer-Snyder metric along the lines of Kegeles 1978 https://doi.org/10.1103/PhysRevD.18.1020 in the axisymmetric case; more modern solutions are mostly considered with numerical relativity) have even less isotropy and homogeneity.

Our universe for all practical purposes is homogeneous and looks isotropic from points far outside galaxy clusters, which is most points. There is no universal "centre" around which anything is orbiting. The mathematical description is the expanding Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is neither axisymmetric nor spherically symmetric, but which is homogeneous and everywhere spatially isotropic.

The Weyl tensor (the tidal part of the Riemann curvature tensor) is very different in the non-vacuum "white hole" solutions from FLRW, with galaxies and galaxy clusters in the former solutions producing a significant redshift-prolation relation for an observer freely-falling away from the white hole. We can also redshift-bucket galaxies and find ourselves (as "Eulerian" observers) collecting sets in planes and annuluses, with a very strongly distinguished set radially through the white hole. It would be attractive to use that set as the cosmological frame, comparable to the attractiveness of the one we use in the standard cosmology.

Time-reversal is interesting. There is a small literature on incoming Vaidya black holes which justifies a tiny handful of papers that treats bulk matter (e.g. galaxy clusters, or the stuff of the Friedmann equations) as the evaporating "shine" on cosmos-originating outgoing Vaiyda white holes. The authors in this area follow some contortions with respect to hiding the central part of such a cosmology (the observable universe must be a truly microscopic wedge in the whole picture in order to hide probably-unphysical observables), and do not appear to have made much progress.

Moreover, this sort of approach approach runs into another problem. "Structure disintegration" at late times (i.e., time reversed structure formation) is tricky, and I think leads you down a path comparable to Einstein-de Sitter 1932, which is now reasonably understood to be a less-than-ideal way of writing down a particular FLRW solution.

Roughly the problem is that you have to evaporate the earliest galaxies (and ultimately the earliest atoms, nuclei, and protons) which is hard to do while preserving the "no drama" condition near a very flat black hole horizon, and even harder to do far from the horizon. Under time-reversed-time-reversal (i.e., back to the white hole spitting out matter in the early universe), how do you recover adiabatic cooling?

We get adiabatic cooling from the metric expansion in FLRW; it's harder to see how BBNS, neutral hydrogen, and the end of the dark ages arise in a white hole solution, at least in the asymptotically flat case. And if you ditch asymptotic flatness for de Sitter (spherical symmetry) or maybe expanding Kasner (axisymmetry), why are you better off than FLRW? Or, back to your point, how does such a universe still look even a little bit like a time-reversed black hole? Accelerated expansion (or accelerated shrinking under time reversal) I think is going to be even harder to lift out of FLRW (with the appropriate value for the cosmological constant) into a cosmology with a central white hole.