> There is a middle ground that is quite common across Europe: a private company that has the government as a shareholder.

Interestingly, Quebec in Canada has been loosely following this model recently and it's working great. Long video but interesting if you like infrastructure stuff:

https://www.youtube.com/watch?v=XlHqqA0onn0&t=532s

TLDR: CDPQ invests on behalf of the Quebec Pension Plan, and has been building infra on behalf of the pension plan across Quebec (and Canada) to great success comparatively.