You're right. Skyscrapers don't achieve any meaningful density because of all the space between them, which offsets their height.
The City of Paris (the classical mid-rise downtown without the La Défense skyscraper CBD), has a population density of around 20000pp/km2, climbing up to 50000pp/km2 in 11th arrondissement.
Downtown Tokyo and London are about the same.
In the developing world, some cities achieve much higher densities with only low rise buildings. Cairo is twice higher at around 40000pp/km2, but granted, the lack of urban planning and mass transit makes it inefficient too, and quite frankly, unlivable.
The only skyscraper district than can rival those is midtown Manhattan, which shies at around 20000km2. However it takes tightly packed skyscrapers to achieve it, at a much lower overall energy efficiency.
Skyscraper districts are typically office buildings, which make population density comparisons a bit misleading. From your example, Midtown Manhattan has few residential buildings vs office buildings, but it gets very packed during the day from commuters and population density numbers don't include any of that. (Manhattan as a whole doubles in population during the day; Midtown Manhattan grows at a much higher multiple.)
Depending on what your definition of "skyscraper" is, they do tend to either add density, increase average unit size, or both. Comparing apples to apples, NYC's densest 5-6 story residential neighborhoods (e.g. East Village) are less dense than the high-rise residential neighborhoods like Upper East Side and Upper West Side. And on top of that, I can tell you that apartments are larger on the UES and UWS.
This Wikipedia article conveniently compares city districts by density. You can see here that Yorkville (part of UES) is extremely dense compared to the densest districts in other cities like Paris, and much denser than community board 3 in Manhattan which is a very dense residential mid-rise neighborhood: https://en.m.wikipedia.org/wiki/List_of_city_districts_by_po...
Note also that Yorkville isn't even full of high-rises. Most of the side streets are mid-rise like east village. It gets the 50% extra density from consistent high-rises along the avenues, and a smattering of high-rises peppering the side streets.
The City of Paris (the classical mid-rise downtown without the La Défense skyscraper CBD), has a population density of around 20000pp/km2, climbing up to 50000pp/km2 in 11th arrondissement.
Downtown Tokyo and London are about the same.
In the developing world, some cities achieve much higher densities with only low rise buildings. Cairo is twice higher at around 40000pp/km2, but granted, the lack of urban planning and mass transit makes it inefficient too, and quite frankly, unlivable.
The only skyscraper district than can rival those is midtown Manhattan, which shies at around 20000km2. However it takes tightly packed skyscrapers to achieve it, at a much lower overall energy efficiency.