Maybe you are confusing what lingua franca means. Yes the literature used Latin but at that period only a tiny fraction of people could read or write. There where many more languages that we even have today spoken by the general populace.

French maybe came close to being a Lingua Franca in the 18-19th century when almost all aristocracy could speak, read and write it but then again the people did not. The Real Lingua Franca was a mix of Italian, French, Spanish and Greek that was used by the merchants of the Mediterranean. Only English has the People Mass to be considered a Lingua Franca for all intents and purposes.

Spanish, too.

Was Latin really the lingua franca, or just the language of liturgy, over that period?

Latin was also the lingua franca of nobles and merchants. Lingua Franca Mediterrania itself, the language spoken by medieval sailors in Italy, Egypt, and nearly all the coastline of Mediterranean is a creole of Italian, Portuguese, and Arabic. To the point that Shakespearean Moors spoke it as "Arabic".

Then, the language of science until the 1900s is Latin. Isaac Newton wrote in Latin. Nearly all scientific papers and books were published in "Neolatin".

Latin was also the language of academia, which gradually separated from the Church starting in the Late Middle Ages.

Both. (Less and less towards the end, of course).