A civilization also includes art. Greek sculpture and architecture remain keystones in Western Civilization. They have been used, adapted, enlarged, and recreated, such as Grand Central Station, in New York City, which is an exact copy of the Baths of Caracalla, based on Greek Motifs.
Civilization also includes the written word, which the Greeks gave us The Republic, by Plato, plays by Aristophanes, Euripides, and others, that are still produced. Almost every single mode of philosophical thought that can be studied today came from the Greeks. Homer is the only author who has so far survived not one, but two dark ages.
Yes the Greeks did not provide everything, but they provided enough to be remembered well, the other major contributor are the Jews, their concept of one God, a set of conduct, and learning has also greatly influenced Western Civilization.
Democracy is a great gift, but it is far from their only one.