How do we understand that the limit $a(t)\to 0$ (where $a(t)$ is the scale factor) lead to a spacetime singularity? Is it by substituting $a(t)$ in the FRW metric and concluding that space makes no sense? But the FRW metric may not be valid right back to the earliest time. In what sense, $a(t)\to 0$ lead to a breakdown of the conventional cosmology?
It's not really about the FLRW coordinates, because you can always introduce a coordinate singularity anywhere you want. Instead it's about the fact that the temperature, density, etc. would diverge as $a(t) \to 0$. So if nothing else intervened, we would have a physical singularity due to the extreme energy densities and gravitational fields.
The singularity is a feature of the FLRW coordinates, and if the FLRW coordinates do not describe the earliest stages of the universe then we cannot conclude it started with a singularity. Indeed, most physicists I know believe that some form of quantum gravity affects will become important near time zero and these will eliminate the singularity. Exactly how this happens will remain unknown until we have a theory of quantum gravity.
(Note that inflation does not invalidate the FLRW metric as a description of the geometry - it just changes the evolution of the FLRW scale factor with time. As long as the conditions of isotropy and homogeneity are maintained the FLRW metric will be a valid description of the geometry.)