> what does an infinite density even mean?

What does t=0 even mean? If we switch to some coordinates where t_0 \def t_now = 0 and t_past \gt 0, as with the scale factor in cosmology, we don't have a special time coordinate at t_past ~ 13.8 Gyr, so we can still think about the stress-energy (and the mechanisms that generate it) and the curvature it sources locally.

We can also think about things under time-reversal:

A comoving test observer in vacuum just free falls without end. Our test observer is a point with no charges (including "active" gravitational charge) and no internal structure, so it feels no tidal effects.

If we have two such observers starting at arbitrarily large space-like separation, they eventually they come close to each other and ultimately end up freely falling on the same trajectory, forever.

If we complicate the test observers from this no-repulsive-interaction picture, we develop pressure (represented in the stress-energy tensor, T^11 T^22 T^33). The local increase in pressure generates local curvature in response, and if the repulsive interaction is finite then eventually the two interacting test observers end up freely falling on the same trajectory, forever. The eventuality is because the background has infinite curvature, and that and the contribution from the self-gravitation of pressure wins out over any finite repulsion.

We can complicate test object interactions further by introducing: repulsive and attractive interactions; extended observers that are composites of these test observers; and even have these objects obey the statistics for bosons and fermions. Same-charged fermions in effect resist being pushed onto the same trajectory, but the resistance is almost certainly finite. Eventually some daughter product free-falls forever.

(We can see some of this in sufficiently massive objects that repulsion or exclusion is overcome, as in the cores of stars or in neutron stars, where internal pressure tends to dominate the local stress-energy.)

What the daughter products of squashed-together Standard Model particles might be is a matter of research, both in terrestrial laboratories and in astrophysical processes (including black holes). Conversely, mirroring the time-reversal thinking above, the Standard Model particles have to freeze out of something hotter and denser that exists at greater lookback time. (cf. electroweak symmetry breaking).

Worse, at a(t_0) ("now") there is a lot of dark matter, and we don't know the details of how that interacts with the Standard Model. Those details will almost certainly matter in the time-reversal picture above.

Consequently we don't know that repulsion/exclusion is finite. Perhaps some crystalline structure develops in the time-reverse picture (or in black holes), and things just accumulate in that. Or perhaps there is a sudden drop in pressure where daughter products fly away from some ultrastrong squashing-together (cf. pair-instability supernovae).

There are quite a few options. And of course the Robertson-Walker background is already just an approximation; the real metric is likely to be very different at high lookback times.

Finally, Carroll has a nice deck (and it's filled with references) at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab...