You're quite right that formation by collapse excludes Schwarzschild.

What you're heading towards is the Vaidya metric -- I don't know of any easy overview of it, but one can think of it in terms of Schwarzschild.

Schwarzschild is a static solution: an eternal, time-independent, everywhere-vacuum, pointlike mass. There is no radiation in Schwarzschild.

Vaidya has the same spherical symmetry, but the central mass is time-dependent. The central mass can radiate away or absorb incoming radiation, but with the condition that the radiation is spherically symmetrical.

Radiation in Vaidya is technical: it is a "null dust" -- it follows null geodesics and does not self-interact, so it shares properties with light ((classical) light rays have no charge, and don't clump; breaking down light rays into particle-like elements leaves each element having no rest mass in its own Local Inertial Frame).

As long as it is sufficiently close to a null dust, Vaidya can model practically any collapse of radiation to a black hole. Unfortunately the spherical symmetry is a hard constraint for the exact solution, and is easily broken by matter self-interactions. However one can certainly play around with numerical approximations to the Vaidya solution, and if one does that enough for a particular family of perturbations of the exact solution, an intuition is likely to develop.

However, a kugelblitz from a collapsing null dust is within the gift of the exact Vaidya solution.

You can take a "swiss cheese" approach to a very early expanding universe filled with radiation (of the technical type above) peppered with Vaidya regions which evolve. Vaidya is time-dependent and can deal with a tapering off of incoming radiation as long as it's always spherically symmetrical, with the result that eventually you have a "cheese" that is an expanding radiation-filled spacetime and "holes" which are vacuoles which asymptote towards Schwarzschild, and around each "hole" a thin-shell Israel junction. The asymptotic behaviour is because the dust crossing the junction into the Vaidya vacuole is (a) cosmologically redshifted within the "cheese" and (b) diluted by the metric expansion of the "cheese". The radiation already inside the junction at early times collapses onto the central mass. The two combine to effectively shut off the incoming radiation, leaving behind a very close approximation of the Schwarzschild vacuole.

(Aside: for massive dusts, one would use a Lemaître-Tolman-Bondi metric instead of Vaidya, and one still runs into problems when breaking the conditions of spherical symmetry and no-self-interactions in the dust. LTB has a couple neat properties which are suitable for physical cosmology "swiss cheese" models if one assumes that the radiation exiting the galaxy-cluster "holes" is negligible -- starlight arriving in our galaxy cluster from distant galaxies likely adds basically nothing to the mass of our cluster as a whole, and our galaxy-cluster isn't losing much weight through its starshine out to infinity; ditto for neutrinos and heavier particles).

In summary, primordial black holes are pretty easy if the very early universe is filled with a homogeneous, isotropic dust -- radiation as a null dust, or some massive dust, or even some combination. In the initial dust one expects Jeans instability, and a power-law distribution for the total masses of the resulting vacuoles. Plenty of small primordial black holes, fewer big ones. What would cut off really small black holes originating along these lines? Unknown. If nothing is found, this type of model loses its attractiveness.

There are several other models for primordial black holes, but they look a lot less like the kugelblitzes you talked about a few comments above.

One other note, although I don't have space to develop it here: we can form black holes from gravitational radiation alone, even in a spacetime with zero matter. Gravitational radiation is not the same as matter radiation in the technical sense above: apart from the mathematical details of which tensors encode it (Riemann vs stress-energy) more physically gravitational radiation strongly self-interacts, so we generally can't treat it as a null dust.

> personal frustration

GR doesn't really come easily to anyone, even (and sometimes especially) with people who are mathematically gifted. Even Einstein and Hilbert struggled with it, and in the last century only a small number of exact solutions -- none of them better than a fair-enough-to-be-useful approximation to astrophysical observations -- have been found. There have been many many many false starts. Consequently one has to develop an understanding of where exact theory starts to diverge from exact observation (and what one can do about it with arcane tricks), and I don't think that's really possible without understanding the exact theory first.

Lastly, I don't know what you're thinking about here:

> the Schwarzschild radius for the mass of the visible universe is bigger than the visible universe

The visible universe is not even slightly approximated by the interior part of Schwarzschild metric. In particular, galaxy clusters are flying apart rather than collapsing to a point. Additionally, there are no apparent tidal stresses on galaxy clusters, even the ones at highest redshift: the Weyl tensor, which essentially encodes tidal stresses, is nothing like a black hole solution (not even under time-reversal wherein we get a "white hole", because we would then see spaghettification in reverse: galaxies evidencing ellipsoidal early galaxy-clusters with later galaxy-clusters becoming markedly rounder).

Event horizons can appear all over the place, including in perfectly flat spacetime (for Rindler observers, for example). There are lots of very-not-like-black-hole-spacetime settings in which there are global event horizons. The salience is in what trajectories radiation and other types of matter take, rather than that the matter-in-the-bulk can be partitioned by horizons that produce Lorentz-contraction observables.

Going back to our swiss-cheese model above: in the far far far future the cheese part is essentially empty of radiation because it has all diluted away with expansion, while the holes are also empty of radiation because it has all fallen onto the central mass. Both empty, highly curved spacetimes, but with very different trajectories for the null dust: flying to infinity versus flying into a point.