The two classic ages of Greece existed first at the very end of the Bronze Age and the then at beginning of the Iron Age. Original swords were made of bronze alloys, not particularly strong, and the leaf shape was the ideal balance for a sword stout enough to slash with, while tapering sharply enough to thrust. Thus, the classic Xiphos.

Now, by the time of Thermopylae and the Persian Wars, a different type of rational came with the adoption of Iron. Leaf swords put more material towards the tip, and thus the center of gravity forwards as well. This makes the sword terrifyingly effective at cutting, much like an axe. As well, like the Xiphos, this sword could be tapered to a point, creating an all-round cut and thrust weapon perfect as a backup to the hoplite's spear. The Greeks called this sword a Khopis, and it gained a fearsome reputation, particularly depicted in the hands of Spartans as a mark of almost villainous effectiveness.

However, putting the center of gravity forward on a sword has diminishing returns on effectiveness the further from the hand it goes, and thus the Khopis was limited in length even though the Greeks had the technical ability to make them longer. This is the rationale behind later swords being straight even if they copied the Xiphos' diamond cross section. Examples would be the Gladius and later Spatha.

A further point is that leaf shaped blades were made elsewhere, just not in Greece as they ceased to be the dominant power. The Falx is a two handed blade that gave the Romans much trouble in Dacia, and a millennium later the falchion in France existed as a cheaper alternative to Arming Swords, and broadened towards the tip for better cutting. Even at the end of the sword's use, the 1798 British Light Cavalry Sabre has a broadened, leaf shape cutting edge towards the tip.

So in summary, early Xiphos styles were the ideal balance between the softness of bronze, demanding more material to weather the shock of blows, and the need for a robust weapon. Later Iron weapons saw the design as an ideal cutting weapon that still retained the ability to thrust, an idea carried on until the end of swordsmanship.

Sources: 1: George Cameron Stone, A Glossary of the Construction, Decoration, and Use of Arms and Armor

2: Tarassuk, Leonid; Blair, Claude (1982). The Complete Encyclopedia of Arms & Weapons: The Most Comprehensive Reference Work Ever Published on Arms and Armour from Prehistoric Times to the Present