Playing Carthage on King/Standard Size/Epic Speed with 3.10.1. I'm doing good, but the Civ's abilities don't seem to have been that overpowering.
I started on a decent location with a fish I could grab early and take some advantage of the early sailing. It's bordering tundra, however, and my capital is facing the "north passage sea" that is usually on the north side of continents. No other good coastal locations to settle next to the capital.
I went Honor (for Opener and Spoils of War, which is a must due to Raging Barbs) and Liberty. Settled my second city in the middle of the continent south of the capital, where some good land was surrounding a lake. Very quickly after that built the third city still more south, on the southern shore of the continent. Then settled a sub-standard location on the tundra on the northern coast to get some resources (iron+whale).
In the late classical, I had gathered enough forces to stomp on my closest neighbor Siam. A militaristic CS ally did most of the work there, 2 gift chariots and 3 gift warriors upgraded to Elephants/Swords nicely. I had to construct the catapults myself, but after I DoW'd, I was surprised how quickly they fell.
So, toughts about the Civ's abilities:
The insta-harbors are for not much. Guess this ability only shines if you find early city locations off-continent and/or further from your capital. But that's very circumstantial. It does relieve the pressure of building roads early, freeing some worker time, but that's minor, especially with Liberty. Maybe it could also help early sea trade, but I didn't find much use in that either.
Elephants are nice, but seemingly nothing special to other UUs. Horses I usually have surplus of, much more so than iron, so not much advantage to not needing the resource. It's a good unit, but I'd rather have Legions, especially with their earlier arrival, and upgrade from warriors, any day.
The starting bireme has been quite useful in the early game to hunt barbarians for the culture/gold (Honor). Certainly much better than the Roman's Liburna. The bireme helped to get to the free settler+workers from Liberty quite much earlier, but with the cost of first taking Honor policies, so probably not that great overall.
The starting Sailing allowed some early fishing, which helped the capital grow. That could be a lot more useful with a more sea resource heavy start, but that is also highly circumstatial.
Purple has helped to relieve some pressure from happiness, and to get some gold after the third city was up. It's proven by far the most useful of Carthage's abilites, at least in this game.
Overall, a early-game heavy civ, as expected, but not overtly so. I think I could have done at least as good with Romans given the same start. With legions taking on Siam earlier and stomping on barbarians even better.
Still, the free tech of Sailing at the start is a bit odd. As Carthage is the only one with a free tech (afaik), it might as well go. You might instead have the cothon available earlier (say, pottery) and/or cheaper.