Whatever is holding back your trains, it's not lack of density. Here are the population densities of some places:
Japan: 327/km^2
England: 434/km^2
US Acela Corridor (Boston-NYC-Philadelphia-Baltimore-DC plus minor cities and outlying areas): 931/km^2 ... nearly 3x Japan!
Yet Japan's rail network is superior in every way to England's, which is in turn superior in every way to the Acela corridor. When I google "acela frequency", the first result is "12 kV 25 Hz AC", which really says it all, doesn't it?
Then there's Switzerland: 207/km^2 and much tougher terrain, yet their train network is Japan-tier. Compare that to California, with 250/km^2 and no serious rail connectivity at all, despite Sacramento-SF-San Jose-Los Angeles-San Diego sitting practically on a straight line. There are only 16 trains per day between San Fransisco and Los Angeles, and it takes over 10 hours. These are two of the most important cities on Earth, it's ridiculous! There should be at least 10 times as many trains, and 1/3rd the journey time.
Look at Texas: Dallas, Fort Worth, Waco, Austin, Houston, San Antonio; jointly they are 20 million people and all exist in a triangle less than 500km to a side. The region is perfect for high-speed rail. There should be hundreds of Shinkansens zooming back and forth across the desert each day. In the manner of Texans, they should be even bigger and even faster. But they don't exist.
The US as a whole is thinly populated, but it still has several extremely wealthy and dynamic mega-regions that have more than enough population density to support world-class high speed rail networks (plural).
You must demand better trains!