You're quite right, Classical Latin represents only a 'slice' through the history of the language, though it was indeed regarded as 'standard' for several centuries, and was resurrected as such in the renaissance. While Classical Latin is what I was lucky to learn at school, at university and since I've specialised in historical linguistics, including medieval Latin. We can trace the development of the language, the ways it changed into proto-Romance and eventually emerged as Italian, Spanish, Portuguese etc. provide evidence for the vernacular dialects, along with a surprising amount of 'non-prestige' written evidence like graffiti, private letters, notes on wax tablets, etc. Historical linguistics, tracing and reconstructing earlier stages of languages, is just as scientific a discipline as evolutionary biology, tracing and reconstructing earlier forms of organisms.
Incidentally, Latin 'lacked punctuation' too, it was mainly written in scriptio continua without even spaces between words, as writing materials were costly. Punctuation as we know it developed mainly as Latin came to be read and (especially) chanted in the church, especially by monks etc. who weren't used to speaking the language. Compare the marks used to guide readers of the Torah and the Quran even today.