Trains would mostly be effective at covering those distances that are currently "puddle jumper" flights. Things which are an hour flight on a slow prop plane, but which take at least twice as long real-time due to airport and security hassles.
There's definitely parts of the US where the population density could easily sustain HSR, though; California from San Diego–Los Angeles–San Francisco is one, and Boston–New York–Philadelphia–Baltimore–Washington is another (quite how far you could stretch that network west is an interesting question, and I don't know enough about current travel trends to comment, certainly Pittsburgh is close enough that travel time-wise it could beat out aviation), as is Miami–West Palm Beach–Orlando for example.
What is interesting is it is likely the case that the North East is the only place where there's a sufficient population density in a larger general area that crosses state lines.
The problems seem to be frequently the fact that nobody wants a train line near them that doesn't stop near them, whereas current aviation routes simply overfly them, and adding in extra stops completely destroys the business case by making them comparatively slow (v. aviation).
No Florida city has a good public transit story. Hell, in the Tampa Bay area, you can't even get from the Tampa International airport to Pinellas county without two hours of buses, because the cross-bay express buses only run for commuters. (Only on weekdays, only during rush-hour times.) So what's the point of a train that goes from Tampa to Miami if it takes me another two hours to get to Clearwater or St Petersburg? That's half of the four hours already to drive direct!
NE Corridor already works well for train obviously. Amtrak has various plans to upgrade service though I don't know the current status of those plans.
It's quite hard to do track maintenance well when there's a train passing by every 5-10 minutes, 24/7. Maintenance windows can become really expensive.