A major part of the space in a city isn't dedicated to residential buildings, but to utilities- roads, parking schools, supermarkets etc etc. Roads is the most major issue with skyscrapers - streets in Manhattan are as wide as highways.

There is also the matter of space between buildings - Skyscrapers usually require some empty space around them, while smaller buildings can be built without any space between them at all (what we call in Israel "Train blocks").

Skyscrapers do allow to cramp more living space overall, yes, but they aren't that much more efficient. Manhattan is still one of the most densely populated places in the world, but cities like Bnei Brak with buildings 4 stories tall can still compete with it, if they house more people per apartment.