Could you explain by the way how you feel each of those civs are only focused on one aspect they do well? Curious to see perception vs various realities on these civs
For all the civs I mentioned:
China - GG's lend themselves to war, obviously, and the CKN is a destructive force to be reckoned with, unless you can get enough damage in it's face to make it move. The Paper Maker is a library that, basically, gives you 3 gold (not exact but a good generalization). Since it's something you want in most cities, that means more money, which is flexible enough, but I feel like this, specifically, was chosen because puppets LOVE to make libraries, making them more profitable quickly.
Arabia - Money money money money. Go grab as many city spots with luxes as possible, sell off duplicates the Bazaar gives, and also get more cash from your low pop cities (you grabbed lots of land for lots of different luxes) connected to your trade routes. They could almost be good for war, but if you decide to let your camel archers loose, you stand to lose at least one trading partner.
Songhai - It's all about momentum. Songhai starts by taking out the barbs, getting an army, taking cities, getting more money, taking capitals, and winning. Their UA screams it. Their UU is a knight that can take cities pretty darn well, adding to it, and their UB is a maintenance free source of both faith and culture which are great to have as you take more cities (so you can spread a warmonger friendly religion to your newly acquired cities and they are faster to pay for the increased culture cost of policies). They're good, but they are really a bulldozer only (compare to Mongolia, for instance, who might take ton of cities but is also generally happy blazing though enemies to the capital as well and has the option of hitting enemies indirectly before war by taking a CS easily...choices).
Greece -They are good at war early and can go for a diplomatic victory easily late. They have a nice balance of when they can take advantage of their UU's and UA, going to war early can produce GG's that can be used for war later, they can do diplomacy later too, as I said, and their UA actually makes culture and space victories easier. They can easily go wide and conquer the world or go tall, taking advantage of the indirect happiness, culture, and military bonuses their UA gives them, all depending on how they want to use it.
Persia - The UU and UB say go wide, but the UA says go tall. In the end it becomes an interesting balance of wide and tall, war and peace, in order to take advantage of a happiness giving UB and one of the best UU's in the game, a warmonger's dream, and souped up golden ages (which also become a warmonger's dream) that are going to require happiness or GA's to activate. Game to game, I don't think anyone makes as many choices based on the current situation as Persia.
Sweden - Their UU's are absolutely made for GG's, since one heals while attacking to make them and the other pairs with them decently well. Add in a UA that gives more GP...so long as you have friends, and each game is interesting. Besides the interesting situation of balancing war and friendships, they, like Greece have many options due to getting a bonus in getting and keeping CS allies. Outside of gold, influence is probably the most flexible thing in the game.