Scientific and medical Latin and Greek are communication tools, so as long as everybody understands the rules and usage within that "dialect" it's all cool. And it's very handy to know things like in botanical Latin, species names like "vulgus" or "sativa" mean "the common species that everyone uses for stuff, you know the one" and not worry about how it was pronounced in Italy in a certain year BC. And people will understand that "semi alba" in that context means a particular color form in an orchid and not "half white" which it would literally translate to.
And Church Latin is a communication tool with its own evolved and carefully developed meanings for words too, and has developed a slightly different pronunciation over time, maybe because a lot of people experienced it primarily as written rather than spoken.
Although there is a stereotype of the uptight teacher who believes there is only one "right" way of speaking, I think most people who teach Latin today try to communicate that it was a living language, like English and every other language. That means that even in a particular year in Rome there would have been different "registers" (more/less formal ways of speaking), dialects, jargon for specialized trades, etc. We can see examples of this, for instance, if we compare a speech by Cicero (highly polished and reworked formal political rhetoric), with plays by Plautus and Terence which were intended to evoke how ordinary people of different backgrounds spoke "off the cuff". But it's true that sometimes people who have invested a lot of time in learning something start to get a feeling of ownership. We call an approach "prescriptivist" when someone wants to teach language "the way it should be", and "descriptivist" is talking about language the way people use it. I started out prescriptivist but am now very descriptivist (notice I ended a sentence with a preposition above, because that whole thing derives from someone's weird attempt to apply Latin grammar rules to English).
We can pick up a lot of clues about pronunciation luckily because there was a high level of literacy, including lots of public inscriptions, and the Empire was very ethnically and linguistically diverse. So we can see how foreign loanwords and names were spelled as they sounded to a Roman ear, and and how a Latin name was written in Greek or Arabic. Also from poetry because there were rules and art to making words fit into verses, although rhyming was not important as it commonly is in English verse. (Rhyming is an important way we know how pronunciation has changed in English, because some words rhymed in Chauncer's poetry, for instance, that don't rhyme today.)
Thank you for coming to my TED talk.