Latin emphatically did not disappear with the fall of the Roman Empire. Classical Latin was preserved not only by the Roman Catholic Church, but also as the lingua franca of all educated discourse in the West into the 18th century (at least!). Initially, this was probably because education was typically provided by the Church, yet its use persisted after the Protestant Reformation and through the Age of Enlightenment before learned folk began to variously adopt French, German, and eventually English.

Classical Latin aside, sermo vulgaris ("the common speech"—"Vulgar Latin") never died so much as it evolved over time into our modern family of Romance languages: French, Spanish, Portuguese, Italian, Occitan, Catalan, Romanian, Dalmatian, Sardinian, …; the list goes on, but I'm sure you get the idea.

I think that, in the context of what I'm seeing throughout this thread, (and to be clear, this isn't directed at you specifically) before we worry that English will establish some sort of linguistic hegemony over the globe, we should first consider the processes by which some language or another comes to dominate international communication in such a way, and likewise the processes by which such an "auxiliary language" comes to be replaced eventually by another. I contend that the main reason English now serves the role once served by Latin is because the Anglosphere currently has much influence on the sociopoliticoeconomicocultural stage, yet it is not currently without rival on this stage. A century from now, it could happen that everyone will be learning Mandarin Chinese or Arabic or something else altogether. Perhaps Esperanto will finally take off.

Returning to Latin for a final thought: throughout medieval Europe this side of Macedonia, all educated people could read, write, and speak Latin, and no doubt they found that these abilities came in handy from time to time. Notwithstanding, it remained in place as a second language, reserved for specific situations and purposes, and they continued to employ their native language in their day-to-day activities—after all, we still have English, Irish, Gaelic, Scots, French, Dutch, German, Polish, Spanish, &c., &c. in the present. These are not the same languages that were spoken in the Middle Ages, nor are they the same languages that were spoken 100 years ago, but, as you pointed out, languages evolve and change.