This is not a bad model: the public retains the land and tracks, and provides them for private enterprises to run trains on, subject to some sort of coordination so you don't have wrecks.
That's similar to how many airports work, how the Interstate Highway System works, etc.
However, for historical reasons it's not how the US system operates, and it would be very difficult for the government -- absent a lot of spending that I doubt the public has the appetite for -- to acquire the land and rights-of-way.
On routes where Amtrak actually controls the underlying trackage, it operates quite well. Where everything goes to shit is when Amtrak operates on others' track.
If we really wanted a good passenger-rail system, what we would need to do is build dedicated passenger rail lines, totally separate from freight rail. The two types of trains are so different today that it doesn't make sense to run them together. High speed passenger trains do best on banked track, and not having freight trains around for them to run into allows you to use lighter rolling stock. Acela trains have been described as "rolling bank vaults" because of the collision protection they have to have -- simply because they're liable to collide head-on with a fully loaded freight train. There's no good way around it, except to stop mixing the two up.
No it doesn’t. Amtrak owns the lines between Baltimore and DC. I used to commute it every day for two years and it was awful. Amtrak struggles to adequately maintain the northeast corridor, because as a result of government-imposed mandates, its operating profits from there go to subsidizing service on lines nobody uses. Similarly, Amtrak is just shitty at logistics and operations—planning around track work, giving updates about delays, keeping trains in working order, etc. Again, they have no mandate to turn a profit (unlike government owned rail lines in Europe) so they run like the DMV, rather than a rail carrier.
That said, Amtrak came into existence because most of the private rail companies had failed or were failing. Had the federal government not stepped in, we'd arguably have no long distance passenger rail service outside the Northeast Corridor (where Amtrak basically makes the money that it then loses on pretty much every other route in the country).
The US rail network is optimized for freight hauling (and it's largely owned by freight railroads and maintained at private expense, and those railroads turn a consistent profit). It also works well -- for its intended purpose.
If you want to move 3000 miles at 45 MPH remarkably cheaply, and you happen to be (or live in) a cargo container, the US rail network is built just for you!