The US highway system did pay for itself out of gas taxes. The highway trust fund was set up for this purpose. See https://en.wikipedia.org/wiki/Highway_Trust_Fund . In recent years the highway trust fund has not been adequate, for two reasons: one, Congress diverted money from the highway trust fund to fund things that were not highways, and two, the value of the tax didn't keep pace with inflation and it hasn't been increased recently.

That plus the clear difference that funding trains would basically mean subsidizing the operations of a few large companies, while anyone can drive on publicly funded roads

Since Amtrak is fully owned by the federal government, why not let them run the trains. That way there’s no subsidizing of a few large companies.

Amtrak is the brand name for the service provided by National Railroad Passenger Corporation, a weird quasi-public, ostensibly for-profit corporation. Its stock is all owned by the government, but it's not supposed to be dependent on subsidies to operate.

"it's not supposed to be dependent on subsidies to operate."

I don't think this reflects the current thinking in Washington about Amtrak. I wasn't able to find its P/L history, but has it ever made a profit? I can't remember hearing that it did.

This is quite similar to Deutsche Bahn, the main German railway operator. It’s a private company but all shares are owned by the federal government. They get subsidies (mostly infrastructure cost) but any profit they make goes back into the federal budget.

Same with airports