The price of Manhattan apartments has almost nothing to do with how desirable it is - other than the fact that desirability is largely a function of the financial opportunities there.
Park City, Utah is expensive and nothing like Manhattan - so is Malibu and Ketchum, Idaho.
San Jose is a large city and much more expensive than NYC (in terms of house prices). Outside of downtown and a few neighborhoods nearby, it's mostly a car-hell.
Los Angeles isn't far behind in terms of prices - it's the second largest city in the US - and the majority of it is less walkable than halfway decent suburbs.
On the flip side, Chicago is mostly walkable and bikable - yet it's substantially cheaper. Does that mean people don't want that? I doubt it.
You can find plenty of counter-examples based on price and population and trying to extrapolate what people want.
People like a lot of different things. Looking at one desirable place isn't going to tell you much.
If you can make an argument that Manhattan has so much financial opportunity BECAUSE it's walkable, bikable, mass-transitable - I'm listening. But keep Chicago in mind. Because it has all of those things - is the second most densely populated city and the third most populous - yet it's basically average price for the country - which makes sense, because it also has average wages and high property taxes...
Why is NYC so much more expensive?
Because people can make more money to spend on more housing in NYC...
NYC metro population grows from 18.3 to 19.8M (8%) [2].
It's harder for NYC to under-build for a century when the population has not grown as much. Chicago isn't exactly a developer's utopia...
[1] https://www.macrotrends.net/cities/22956/chicago/population
[2] https://www.macrotrends.net/states/new-york/population
If you isolate just the city - NYC does grow from 7.8M to 8.8M (12%). Manhattan goes from 1.54M to 1.69M (9%).
Chicago's downtown population growth is >40%.
In any case, though, the original question is whether population density actually makes a place less desirable. The mere fact that Chicago is not cheap (not simply inexpensive, but actually cheap) contradicts the claim that population density is something people don't want.
They don't want density for the sake of density.
NYC has better opportunities - that's why it's denser. It's not denser because it's more desirable, and then it just so happens that it also has opportunity.
This is true for Shenzhen and Beijing and Mexico City and Paris and London and Mumbai and every major city.
NYC is located at the mouth of the Hudson and one end of Long Island sound, two relatively sheltered waterways. For hundreds of years this facilitated huge amounts of resource extraction from the region. Upon this resource extraction economy resource processing was built. Naturally it was located where the labor was the the products flowed through, the port. From there industrialization and an industrialized economy way built upon the resource processing. Through little other than historical luck of geography of the industries (that really took hold in NYC was finance. It has been that finance industry that has carried NYC through the transition to a service economy.
So basically, NYC is valuable today because it was yesterday and if you follow the recursion back to the base case you find natural resources.
While the natural resources are now gone or irrelevant they were the startup capital for the NYC we have today. There is no inherent value to a mass of people unless there is some productive activity for them to engage in. NYC got lucky and developed a finance industry. See Detroit for what happens when you don't make the transition.
People can't finance land. You finance housing. Sqft / sqft for housing, Manhattan is not much more expensive. The land is much more expensive BECAUSE you can build much more housing.
If New York could similarly encourage speedier housing construction and replacement, as well as effectively building a similarly expansive and reliable transit network, they could achieve the exact same results as Tokyo. (Note: Even though Japan has had periods of economic stagnation and low birth rates, Tokyo itself has consistently grown in population with few exceptions as Japan has become more urbanized.)
If the suburbs are so great for housing supply, why do suburban Boston/San Jose/Austin/Denver homes cost so much? Why are Chicago's most prestigious zip codes more affordable than living an hour outside of San Francisco?
The truth is that the places with high housing prices don't have high prices based on whether they're suburban, rural, or urban, they have high prices when demand exceeds supply. It's that simple.