We practically used to, until cars encouraged development to spread out.
> You have to completely rethink density and development patterns to get to a point where most origins and destinations are near stations.
In other words, go back to what the world was like before cars.
> Public transit has a wickedly hard problem to solve
But not impossible. European cities seem to have a less difficult time.
European cities had the good fortune to lock in a healthy level of density as "neighborhood character" thousands of years ago, so that the same sorts of structures Americans read as the rapacious excesses of a new era of unfettered capitalism and runaway inequality (i.e. 5-story apartment buildings) are instead seen as historic. They also aren't in a position of having to unwind the "mistake" of certain neighborhood communities existing.
>In other words, go back to what the world was like before cars.
The places we would invade to "go back" to, and the places we would abandon to "go back" from, are now all occupied by humans with voting rights and a preference for the status quo. Not to mention that the population is simply larger, and more concentrated in metro areas. The city centers abandoned during suburban flight were quickly claimed by low-income people of color, and also a bunch of extra people were born in (or migrated from farmland and small towns to) suburbia.
>We practically used to, until cars encouraged development to spread out.
The tenements of pre-car industrial cities were so traumatic that we still have the scar tissue today, in the form of wildly overbearing zoning and building codes. The Progressive movement's obsession with making sure buildings have enough light and air and open space, which holds back San Francisco from even European-level density today, is rooted in the outrage felt towards living conditions 100+ years ago, which the car saved us from.
Obviously there's a healthy middle ground and it's frustrating that we don't take it, but our culture has the memory of what it was like to live too close together, and quite likes the ability to spread out.