general relativity - Interpretation of a singular metric

I'm interested to find out if we can say anything useful about spacetime at the singularity in the FLRW metric that occurs at $t = 0$.

If I understand correctly, the FLRW spacetime is a combination of the manifold $\mathbb R^{3,1}$ and the FLRW metric $g$. The metric $g$ has a singularity at $t = 0$ because at that point the proper distance between every pair of spacetime points is zero. Presumably though, however the metric behaves, the manifold remains $\mathbb R^{3,1}$ so we still have a collection (a set?) of points in the manifold. It's just that we can no longer calculate distance between them. Is this a fair comment, or am I being overly optimistic in thinking we can say anything about the spacetime?

I vaguely recall reading that the singularity is considered to be not part of the manifold, so the points with $t = 0$ simply don't exist, though I think this was said about the singularity in the Schwarzschild metric and whether it applies to all singular metrics I don't know.

To try to focus my rather vague question a bit, I'm thinking about a comment made to my question Did the Big Bang happen at a point?. The comment was roughly: yes the Big Bang did happen at a point because at $t = 0$ all points were just one point. If my musings above are correct the comment is untrue because even when the metric is singular we still have the manifold $\mathbb R^{3,1}$ with an infinite number of distinct points. If my memory is correct that the points with $t = 0$ are not part of the manifold then we cannot say the Big Bang happened at a point, or the opposite, because we cannot say anything about the Big Bang at all.

From the purely mathematical perspective I'd be interested to know what if anything can be said about the spacetime at $t = 0$ even if it has no physical relevance.