From the question:
You and I speak "Common" – it's called English. But this is the result
of the recent globalization made possible with the advent of the
This is incorrect. It is the result of Imperial Conquest, and that, I think, is the real heart of the matter.
If you look at Common as an imperial language - an official language of government - then yes, human kingdoms at war and "non-humans" (and let's be frank, that notion is grounded firmly in Tolkein's quaint victorian notions of race) would both speak it and their own language, even if the empire is a fading memory.
Take, for instance, not a continent, but two subcontinents - India and Europe.
In medeval Europe, if you knew Latin, you could generally find someone in town who knew it as well - clergyman or clerk - and you could fake your way through a conversation in a place that spoke a romance language, if you hit your language rolls. So, Latin as "Common" would still require a player to sink some stats in languages if they want to talk to random villagers.
In modern India, you have the language of Empire, English - if you do business or deal with the law or government, you speak it. You also have the language of faith - Hindi - which even non-hindus learn in order to communicate with others in the community. Then you have twenty one "mother tongues" - languages learned from your mother, this is the official language of the place where you live. Of course, there are even more unofficial "mother tongues", the language of your ethnicity, of your social caste, of your particular village that's different than the province's.
So, you would have an imperial language or two, "Common1, Common2", and some NPCs might know one better than other NPCs, but most everyone would know a smattering of either. Mother Tongues can be then broken down by race (ethnic-centric language) and alignment (caste-centric language).
So the way to run the campaign is to make the players roll language skill to speak common, to see if they can actually communicate. Knowing another "Mother Tongue" fantasy language, elvish or evil, improves the ability to talk to those NPC's that might also know them, even in passing.
Jaques Cousteau has a story of his wife, French, trying to hail a Greek captain on a nearby yacht, and both parties are attempting to say hello in every language they know - and tho France and Greece are only a few hundred miles apart, they wind up speaking in Japanese! In this way, learning languages should improve a character's ability to speak with others generally.
See also, Lingua Franca a "third language" that people who don't know each other's languages communicate in, and poorly.
Everyone knows Common. No-one knows common very well.