> Why is what Latin did relevant for French in the 21st century?
Scholars have always gone back to Latin to influence modern language. Most technological, scientific, and plenty of engineering terms, for instance, derive either from Latin or Greek.
Modern French didn't 'branch' from Latin so much as the Latin dialects in the areas that have since become modern France evolved from Latin into what it has since become. Therefore, going back to classical and vulgar Latin tells us a lot about how the language has evolved, what features still provide valuable meaning, allowing people to guess what a word means based on their existing base knowledge of the language.
Doubly so considering the French curriculum remains relatively old-fashioned compared to the Anglophone world, such that explicitly teaching grammatical concepts at a very young age has not yet gone out of fashion in France and other Francophone regions and nations), so that the general population has a firmer understanding of the derivation of terms, the meanings of prefixes, suffixes, and so on.
Not only that, it allows the modern French language to maintain a sense of consistency by going back to the roots and origins of words in order to determine how best to coin new words without causing needless difficulty, having to create exceptions where unnecessary. Hello, English language, especially in the past hundred or so years with 'electrocution' where the most important part of one of its components, 'execution', is entirely missing (such that it's not clear just from looking at 'electrocution' that being killed, 'executed', is a key part of the word).