How so? The very fact that someone makes a profit means that the end consumer is paying more than cost price.
If I pay £10 for something that costs £5. The organisation that just pockets the difference is less efficient from my perspective than the one that either reduces my next bill by £5 or gives me an extra £5 worth of service.
The point is that even if a private sector company operates with perfect efficiency, there is a built-in inefficiency from the consumer's perspective because investors need to make money on the deal.
Efficiency is not part of the definition of public sector or private sector, so clearly your statement is false. It's inconsistent with the rest of your comment because you wrote "It's not a guarantee that a particular public sector project or department", which could not be case if it were true by definition. Your argument that public sectors, depending on circumstance, tend to be more efficient is quite different from your assertion that they are more efficient by definition.
What I mean is that all else being equal, public is more efficient than private. The same team doing the same task with the same information, constraints, and money will be able to provide better value to the consumer in the public eector than the private.
The caveat is there because a handful of strategically placed idiots or geniues could easily make more of a difference than the fact that all the money stays within the system.
This is not a reasonable position, nor is it consistent with the rest of your comment.