The situation is more tricky than just verification because there are so many parameters and assumptions involved in various models, for one we don't even know for sure whether inflation really happened or not as there are other viable alternatives that can explain observations without inflation though these are generally far less developed. As for cosmic censorship one thing to remember is that we know General relativity as it is currently developed is incomplete so it is very well possible perhaps even probable that within a more complete picture of general relativity the singularities may disappear altogether.
The real questions should be what is the simplest model which can resolve these apparent issues and would that have any observable consequences.
For example one of the things inflation was initially proposed to explain was the apparent homogeneity of the CMB, however the "isotropic CMB" is only obtained under the kinematic dipole hypothesis which has been assumed to be true. There was an experimental test which was devised that would be able to verify or falsify this assumption by constructing a dipole from cosmologically distant sources and checking if it is the same magnitude and direction as the CMB dipole which should be the case if the dipole is kinematic, however it was realized you need millions of sources in order to construct a dipole of sufficient accuracy to test this to any degree of confidence. Keep in mind most studies of cosmological objects generally have only had tens to hundreds of thousands of objects at most which means even relatively famous results like the "discovery" of dark energy(which really is the discovery that the expansion of the universe is accelerating dark energy is an interpretation to explain that observation) was only just above 3 sigma significance.
This was recently finally tested with a sample size above a million sources in "A test of the cosmological principal with quasars" Nathan J Secrest et al 2021 APJL 908 L51
https://iopscience.iop.org/article/10.3847/2041-8213/abdd40
Using a final sample of 1.36 million cosmologically distant quasars after extracting known sources of contamination they found a dipole which has a 4.9 sigma discrepancy with the CMB which if verified is more than enough to falsify the assumption that the CMB dipole is purely kinematic.
If there is a cosmological component to the dipole then it isn't anomalously homogeneous so one of the major lines of evidence inflation was proposed to explain vanishes leaving only the apparent lack of magnetic mono poles as a mystery which needs inflation or something similar to explain.
In light of this and related work studying the full Inhomogenous and anisotropic Einstein field equations I'm increasingly skeptical that inflation is even needed at all.
Also with the recent discovery of the gravitational path integral and its ability to solve the information paradox without requiring supersymmetry or string theory gives me hope we may see quantum gravity arise sooner than later with far less necessary assumptions.