The argument about Europe having greater densities does not hold up - add up the population of the stops before the first TGV line (Paris-Lyon) and the stops after the TGV and it's pretty clear that France's population outside of Paris is pretty comparable to U.S. populations between major regional city pairs (Dallas-Austin, San Francisco-Sacramento, Chicago-St. Louis).
There are about 300,000 people in the cities served by the old French rail lines - an amount exceeded by the population between a number of the proposed U.S. high speed rail city pairs. The advantages of the train aren't just speed, though - it's convenience (city center to city center service), multiple intermediate city pairs (Springfield-Milwaukee or Madison-O'Hare, for instance), lower environmental impacts, lower operating costs, and lower spatial requirements (a 2 way high speed rail line is about the same width as a 2.5 lanes of a motorway but can sustain significantly higher speeds, greater numbers of people traveling at the same time, and manage congestion more effectively; also a track gate can be reused in less than 5 minutes rather than up to 40 minutes or more for an airport gate). The main benefits are the increase in intercity trade along the high speed rail lines; directly linking business, government, and research centers; and the increase in businesses in sustainable city centers served by effective transit.
Airlines to intermediate cities are often heavily subsidized by federal if not state and local governments and multi-billion dollar airports with dozens of years of long term bonds to not just pay for construction but also to mitigate the noise and traffic congestion that surround them. These costs are hidden from airline balance sheets and airline stockholders by the responsible governments and airport authorities. Airlines quickly become unprofitable as their incremental costs (non-airport terminal capital, non-runway capital, non-air traffic controller fees, non-security) increase due to contract negotiations, increases in fuel, or increases in maintenance costs - federal and state government often step into bail out the airlines to maintain competition.