Amtrak receives well more than $1 billion a year in federal subsidies and still loses millions of dollars per year. It is private only in the broadest sense of the word.

Amtrak doesn't own much right of way.[1] Only about 700 miles. There are about 140,000 miles of track in the US. Nearly all of it in private hands.

In France, the railroads are nationalized. A new high speed line doesn't incur the similar costs for right of way acquisitions.

[1]: http://blog.amtrak.com/2019/05/why-are-amtrak-trains-delayed...

Amtrak doesn't own most of its rails. They have to negotiate with, e. g. BNSF and Union Pacific to use their tracks in most of the country. This often has led to limits on services and schedules, because they don't want to repackage their profitable freight offerings, or upgrade trackage sufficient for 100kph freight trains but not 200kph+ modern passenger ones.

But the point above is that it doesn’t get any of the “benefits” of being public, even if it is a de-facto public entity.

The new TGV are all new right of ways for which they had to buy the lands, except in the suburbs where they go through upgraded original tracks that are shared with suburban trains.