Even if you built train lines everywhere outside urban areas, any stop you step off at would still require a 20 minute drive to get to where you are going.
Sweden - Stockholm (5260/km2), Gothenburg (2936/km2), Malmo (4641/km2) Tennesee - Nashville (557/km2), Memphis (777/km2), Knoxville (782/km2)
Averages lie.
The density map of the US already resembles the US highway map. Many of the few pockets that don't are explainable with old passenger train maps overlaid on top of that.
Just adding passenger train-only routes on top of existing interstate highways, stopping only at existing exits, would go a long way to service a lot of the US population. It would also presumably spur more walkability efforts at a number of those exits if it was also a passenger train stop.
Also population per km^2 can be incredibly misleading, because it's an average. One example could have the population far more evenly spread across the land area, and another could have a dense urban core surrounded by farmland or national parks.
Florida has a slightly higher population than Denmark, but as an example, one of it's largest cities, Orlando, has a population density of ~283 people / km2, while Copenhagen has a population density of 6800 people / km2.
I have lived in Florida all my life. It is sadly the definition of sprawl. There is no way to build enough rail that would serve the average Floridian for day to day transport.
The overlapping network effects OP mentioned of how modern metropolises were built, combined with modern bureaucracy, make widespread passenger rail an exceedingly difficult political and economic proposal - far more difficult than it would have been 100 or even 50 years ago.
I'm sure others have mentioned this here, but look at how high-speed rail in California is faring for a good example of the political difficulty involved. The legal hurdles, environmental reports, back-and-forth bargaining between varying levels of government, the NGOs that need to be called in and paid a lot... it's not at all easy. And that's before you get to the structural mismatch between widespread rail transport and all the network effects that led to car culture.
Perhaps Tennessee can fare better due to it being a comparatively very conservative state (which currently codes as having less regulatory burden). But I'm not holding out hope.