I don’t think there’s a consensus on how to fix non-European civilizations, but this is my suggestion. 
Asian civilizations.
Remove the wonder system and the export resource.
Give it the age up system that African civilizations have.
Replace the Japanese shrines with royal houses without any special effects.
Change the monks/heroes to a single mounted scout/hero.
All Asian settlers cost food and all units change their stats, costs, and functions to be more similar to European units. (this is for balance)
Give visually unique factories to each civilization.
Change the appearance of the ninja to be more historically correct.
Remove the Chinese recruitment system and make it more similar to the Russian one, both in units and villagers.
Native civilizations.
Remove the plaza/bonfire and change it to a proper temple/shaman house.
Remove the magic system and turn those rites into technologies with reasonable stats. (which could be researched at the temple and are equivalent to the European arsenal)
Give them the age up system that post colonial civilizations have. (Territory, allies or the war council itself, I don’t care, just give them 2 cards for each age)
Give artillery, cavalry and gunpowder weapons to the Incas, plus they should be able to have an advanced arsenal. (through a card and if the arsenal is European)
All these civilizations should have economic improvements from the industrial age that allow them to compete against civilizations with access to factories.
More variety of artillery for native units that have access to them and more variety of captured artillery for those that historically did not have access to artillery. (this is for pure balance)
All units have to be equivalent in cost, stats and function to European units. (obviously for game balance)
For both groups.
If a civilization can have a revolution, then it should work exactly like the European revolutions and should be on par with the Imperial revolutions. (French, Texan, Mayan, and Californian Revolutions)
No civilization should lack basic units for the game period and for the base game.
All civilizations should be able to excel at something, but through the age-up system and cards, not through arbitrary mechanics.
If a civilization lacks something historically, it can be compensated for with new or old mercenaries, there is no need to give super units.
I realize that my opinion is not shared by almost anyone on the forum, but I think this would greatly improve these civilizations and make it easier to balance them, while also making it easier for more people to use them.