Ok, we have to set up the definitions.
There are two versions of the phalanx. The old hoplite phalanx and the newer Macedonian phalanx. The only thing they have in common is the name and their relative unflexibility. And neither was invented by the Thebans, they still used the old hoplite phalanx when they faced off with Phillip and later with Alexander. The inventors have been said to be Pheidon of Argos (hoplite phalanx) and Iphicrates and/or Phillip of Macedon (Macedonain Phalanx).
The hoplite phalanx was a defensive construction where each man got protection from his buddies to the right and left. Especially to the right as his own shield was centered on his left elbow, making a bit of his right side open to attack, but at the same time the shield would also stretch out to the left and protect the next guy's right side. The weapons carried was the spear and sword. The spear was 'short' about 2.5 meters and carried overhand (some, it seems, did carry it underhand).
The Macedonain phalanx is the most like the one we see in the game, but it was based on an offensive outline where the men were protected by pikepoints rather than by shields. This demanded longer spears (pikes) and smaller shields strapped to the forearm. The pike was wielded twohanded and low.
The reason the hoplites use a onehanded underhand grip is because CA came to the conclusion that the overhand style was problematic (if you watch TC Adrianople you will see the Auxiliaries use the overhand style), the same is true for the twohanded grip of the Macedonian phalanx.
But to answer the original question. Yes the Spartans adopted the Macedonian phalanx, but much later than most others. It was around 220 BC.
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