"Just build up" is not a simple, problem-free solution. The physical (gas, water, waste removal, transit capacity, etc.) social infrastructure (schools, police, hospitals, etc.) and more intangible desirables of urban planning (noise levels, green space, continuity of neighborhood design) don't just magically expand to fit the density. Maintaining the quality of life that will keep property values at least relatively stable (and thereby the tax valuations to maintain everything) does not necessarily rise linearly with the number of people you add, and when you have preexisting inefficiencies to work with when the original design wasn't done with the record urban population rates we're seeing now in a modern standard. There are plenty of urban slums around the world where people have space to live, if you can call that living. Now that's an extreme, but if you want to forge ahead and sacrifice some quality of life for everybody by upzoning, don't forget that the preexisting tax base also usually has the greater resources to move.

That's not to say we should never build, or find innovations and consider incentives to make urban living more accessible, but let's not pretend that easy answers without significant costs that others will tolerate idly while legislation eats into their self-interest.