They could also be completely user-funded, if politicians weren't so squeamish about the gas tax and other user fee-like schemes. (Albeit somewhat unfairly, because folks using "profitable" Massachusetts interstate highways would be subsidizing vastly less-traveled roads in Montana.)

Amtrak, outside the NE corridor[1], could not possibly be purely user-funded. Increasing fares would decrease ridership significantly enough to offset any added revenue. Whatever fare rates would maximize revenue couldn't pay for the system.

[1] And I'm skeptical about this, too: Gas taxes pay for the lion's share of capital expenditures to build and repair our interstate highways. Reports like this about Amtrak's "profitability" are limited to operating costs, which are all you need so long as your tracks and trains last forever.

That's because relatively no one takes trains because they are too slow. There are clearly major population areas where fast trains would be quite popular.

Sure. It's just that very few of them are between the 10 largest US cities:

https://news.ycombinator.com/item?id=8713324

Very few train riders are large-city-to-large-city passengers. The most successful train lines may terminate in a large city, but it is the density of the route that matters, not the termini. There are plenty of very successful routes in the world that are between two small cities, or only with 1 large terminus. The LGV Mediternnanee, from a demographic standpoint, isn't much different than a line from Indianapolis to Chicago line, and the Nagano Shinkansen line not much different from a Seattle to Portland line. And yet both of those lines are extremely profitable.

This is mostly due to simple competition...the existence of passenger airliners, regardless of how competitive the train or airline offerings are, reduces market share. I guarantee you that the most profitable ticket sales on California's HSR fairy tale will be between (LA|SF) and (Fresno|Bakersfield|Modesto). Not between LA and SF.