"Ancient Greek" covers a long period but yes as a general rule Greeks often besieged cities as part of warfare and when they did they tended to circumvallate (build a wall around the city and stop access in or out) the city and wait until their food ran out / a faction inside took control and opened the gate.

Hoplites were heavy infantry used in massed formations (a phalanx). They carried armour, a helmet, a shield and a spear and sword and were meant to hold the line in an infantry clash. Hoplites were totally unsuited to assaulting a city due to their relative immobility, exposure on their right side and the extreme difficulty of climbing while holding a spear and a 8kg shield.

Nevertheless if we expand your question to ask why did the Greeks not have a dedicated city assaulting unit we can identify certain features that appear relevant.

Assaults on walled cities were practice by the Assyrians and Persians. So the concept existed. However unlike the Greek polis, these were monarchical empires that could draw on orders of magnitude greater resources than any single Greeks polis. A polis having a citizen army may have been reluctant to waste their limited manpower and the inevitable casualties in a direct assault on a city's walls.

The Greek approach was to trade time for casualties. They would surround a city with a wall and starve the inhabitants into surrendering over weeks or months.

In this the Greeks were extremely effective. In the 50 year period between the Persian and Peloponnesian Wars, there were 30 sieges recorded and 27 were successful.

Probably significantly the majority of those 30 sieges were led by the Delian League, that is could draw on the resources of several polis, and therefore could sustain and provide for an army in the field for an extended period.