Yes, the methods can be used to reconstruct Latin (and other known languages). However, there are some practical difficulties, specifically that reconstruction results in Proto-Romance, which is similar to Vulgar Latin (spoken), not Classical Latin (written), so the connection is not direct. For example, *v is reconstructed as /v/, rather than as /w/ as it was pronounced in Classical Latin, because that innovation is shared throughout Romance. Doing a completely unbiased reconstruction of Latin is also difficult because most historical linguists know at least something about Latin, so it's hard to really use this as a blind verification of the methodology. But, yes, overall it works.

As for checking accuracy, one interesting case is how the laryngeal/glottalic theory of PIE was proposed without any direct evidence before the discovery of Hittite (as being related to PIE at least), which somewhat verified the existence of distinct H-like sounds. (The details are still somewhat complicated/controversial, but that is an example of new evidence being discovered in support of an existing hypothesis.)

Overall, the confidence in the results of comparative reconstruction are all relative. It is almost completely certain that the idea of PIE is correct-- that there is some common ancestor for all of those languages. But it is much less certain exactly where it was spoken, by who, or when. And we likewise have a pretty good idea of many of the phonemes but much less of an idea (if any) about the allophones and actual specific pronunciation of the phonemes. Really, all that reconstruction gives us is a mapping between modern languages that shows connections, and then we imagine that as a sort of artificial stage in the proto-language. Reconstructions for PIE, for example, range over maybe 2000 years, so it's really not the case that there even is a literal "Proto-Indo-European" to reconstruct. If we went back in time with a time machine and tried to speak with them, what exactly would it sound like? Well, that's tricky. When? But it is likely that we are right about, in general, most of the phoneme inventory and that would correspond to something at that point in time. So the reconstruction is something like an outline of the language, and indeed some aspects are probably quite reliable, while others are much less so.

So, in the end, it depends. The whole methodology should certainly not be abandoned. But many of the details are far from certain.