Do they ever really go away? Seems like most games still have plenty of fog at least into the industrial era. Curious to see if VI's less harsh approach to expansion actually makes barbs an early game only thing.
In Civ IV, for all practical effects, the barbs did eventually go away; and similarly for all previous Civ titles. (SMAC excluded due the equivalent being created from nowhere as a result of the equivalent of pollution) The Civ II manual in fact noted that eventually there will be no new barb camps forming.
Maps with a lot of hostile land terrain did result in them sticking around a lot longer than normal maps though.
For Civ VI, timing is going to be land mass type specific assuming default number of major civs & city states for a given map size.
On a pangaea, unless there's a region especially harsh within it, I expect all barb camps to be permanently cleared from the main land mass (baring a massive slash, burn, and raze cities campaign that doesn't found any new ones) by the middle ages.
For continental, the same thing should occur slightly earlier on landmasses in which major civilizations started.
Islands in which no body started though that also require crossing ocean to reach are likely to have barbs in the age of discovery.
Playing on something like a "Terra script" (all major players start on the larger main landmass and no one other than city states on the smaller land mass and not accessible without crossing ocean), will likely have carpet of doom barbarians by the time it's discovered.
I would tend to agree that if there's any vanilla civ in Civ VI that can recruit barbs like Germany / Ottomans in Civ V that of the remaining civs, Rome is most likely. But this mechanic may now be restricted to Apostles only (for any civ but only ones with that promotion).