Tyrvos
Cu Chulainn
Brazil's culturally European (you can point out that it's "close to Latin American" countries in a way, say, America is not - but since every Latin American country is of European origin this is immaterial). Byzantium's territorial extent is almost identical to that of the Ottomans - if we're treating the Ottomans as a European civ as well, that gives us two more medieval European civs.
Era by itself is fairly coarse; it's true that most European civs are Renaissance, but these are mostly late Renaissance (lots of Lancer replacements, Caroleans replacing Riflemen etc.) I had, indeed, forgotten the Celts (deservedly so, in my view) - you're right that I neglected Germany (also, technically, France, who now have only a medieval UU, however they don't have an early-game playstyle).
At the time of the Byzantine Empire, the Southern Balkans and the Anatolian peninsula were almost purely culturally greek (from Alexander, Seleucid Empire, etc.) The Turks then invaded and established the Ottoman Empire. There was an 1000 year difference between the start of the Byzantine Empire and the start of the Ottoman Empire. So even though the Byzantine Empire was largely Greek, by the time it fell, different cultures of people had migrated throughout (Even Egypt was largely Greek at the start). There were still some Greeks in the Anatolian peninsula, but they over time were pushed out by the Turks. This is why the Ottoman Empire is not European, they were Turkish nomads that settled in modern day turkey. Ottomans have a special musket and lancer, neither are medieval and they did not reach their peak until after the fall of Constantinople, which marked the end of the medieval era.
There's something like 5 European UU's in the medieval era. That isn't too bad, and I'm not counting UB's or UI's.
Out of the European Civs, two (Spain, France) have musket UU's who are early renaissance. England has the Frigate, Portugal the Caravel, the Dutch the Sea Beggar. I wouldn't discount the Lancer as late-renaissance, since each era is only two rows so if you focus to it you'd still get their early, so that includes the Swedish and Polish. None of these are late game, really.
You have a lot more knowledge about Asia, so I believe you for what your saying there. The makers of this game, however, do pick what they consider their most influential times though. It's like with Greece, they were more than just Ancient Greece, but they picked their most influential era.