I have a lot of experience taking trains all over Europe. In almost no case was a train cheaper than flying once I included transportation to the airport. The few times a train costs more the difference is marginal and you'd end up flying Ryanair or Easyjet --- no thanks.
> And then the profitability of short distance routes is a direct product of the density of population.
Not a problem in the Northeast in the US. Only Amtrak can fail to offer a service that works everywhere else in the world.