Essentially they did, it was called Latin. It was modernised in its pronunciation and even developed new vocabulary and sayings as it continues to be used as the language of the church, the language of education, and the language of diplomacy for centuries. Modern french spelling even reflects this with some of the X's for example stemming from shorthand for Latin. The expectation that educated people could speak Latin lasted untill quite recently, and hundreds of thousands of students still study the language each year.

I can't say exactly when this started to change, but the Reformation and printing presses definitely played a role, with the massive success of vernacular scripture and corresponding pushes for education of the masses. At some point french became the language of diplomacy, but Latin remained the language of the Catholic Church through to the present to varying degrees, and scientists still use a tonne of Latin in biology and anatomy. The rise of new empires that rivalled and even surpassed the Roman Empire really seems to have been the biggest pressure, resulting in new vernaculars becoming global lingua francas, like Spanish, French, Russian, and English.