If you compare a state run/command economy (such as the old USSR) there is no confusion that it does not generate value/income/wealth even without a private sector, the issue is its generally accepted it did so less efficiently.
There are lots of times when private has been inefficient, esp when it comes to infrastructure (this thread!)
The private railways that marked the initial railway boom in this country often built multiple redundant competing lines. Eventually many of these companies went bust as their debts mounted.
Today, the fiber rollout where multiple telcos are overbuilding the same areas where they are chasing the same customers, whilst ignoring other areas they all consider non-viable. Overbuilt 4x in some places! And then each different company erecting their own telegraph poles, as they refuse to share each others (only BT is compelled to share).
Again, we're seeing multiple companies enter administration, mergers and acquisitions, etc.
The whole goldrush/boom then bust cycle is itself surely inefficient, compared to an organised, planned rollout by a publicly owned body.
I'm not sure where the general idea of private sector = efficient comes from, but it's been made a mockery many times over - esp wrt to infrastructure.
And with the current trend of mega mergers which shows no sign of stopping, we may simply be headed for an period where there is only one player in many markets. Will be fun to see how "efficient" a private sector monopoly is. I mean, many US states have monopoly telcos, and they hate them.