I think people often confuse the axiom...

Stated more correctly, smaller, more focused, organizations outperform larger, more bureaucratic(more management) organizations for particular tasks.

It's not that governments are outperformed by the private sector every time for everything (some things require size and longevity). Conceptually, it has nothing to do with whether an organization is private vs public. It's about incentive, regulation, and consequence horizons.

Incentives
Generally, private organization's incentives (if tied appropriately to the desired outcome) are more aligned and offer higher rewards. Government organizations, if completely transparent, have a hard time justifying high rewards to their constituents. All other things being equals, people prefer higher rewards. So, the talent tends to accumulate in the private organizations. There are some exceptions to this (political influence), but I'm generalizing. :)

Regulation
The smaller the organization, the fewer rules. Not only do laws tend to work this way, but internal organizational rules work this way as well.

As organizations grow, managers try to scale their ability to influence direction by legislating organizational rules. As more and more of these rules come into play, the overhead and unintended consequences of these rules grow. Since these managers are unable to directly participate in the execution of the tasks (the reason they created the rules at the start), they are slow in detecting changes that should require rule changes.

The overhead and lack of agility created by both size and expanded regulation(rules), makes large organizations far less efficient than smaller organizations. Unfortunately, governments have disproportionally larger sets of rules. Not only is their ruleset larger, and their employment not aligned to performance, but they also have unionization that has a differing set of goals. This creates even more operational friction making it very difficult to allow people to do what they do best. Tragically, this makes efficiency and productivity in large government organizations nearly impossible.

Consequence Horizons
Smaller organizations tend to have better alignment between their existence and their ability to execute. This is because they are paid for execution and are typically unable to structure longer term contracts. As a result, if the people paying the bills are not happy, they cease to exist quite quickly.

As organizations grow in size, they start to manage for risk avoidance and not for execution. The larger they get, the more they are able to influence the market to avoid direct competition, to change the criteria by which they are judged, and to structure contracts such that they are far more stable over longer periods. This 'cushion' allows these organization to ignore or tolerate consequences to a higher degree. As a result, they are less sensitive to market judgements about their efficiency, productivity, or ability to execute. They loose sight of that as a goal and start to engage in higher order goals (happiness, employee well being, brand, reputation, influence, etc). These are not bad necessarily, it just makes them less efficient at particular tasks.

In particular, large government organizations are disproportionally disassociated from consequence. This is directly related to election cycles, politics (spinning the message to create success out of failures), and constantly getting inexperienced leadership. The first two are points are obvious, but let me explain the last. New politicians are elected and they appoint the people that they feel are best to run these large organizations. Typically, neither the politician or the person appointed has never spent a day working in that organization or executing the tasks for which they are responsible. With large organizations, it takes truly special managers to make an organization efficient at something you have never done yourself. This is even more difficult when you are not answering to your boss for your ability to execute, but instead are answering based on the ebb and flow of the political landscape.

In general, my point is, that you are right. Large private organizations do not have a significant advantage over large public organizations. That advantage can be widened based on political systems. However, the real advantage that people reference when making this argument is the enormous advantage small to medium private organizations "enjoy".

Understand that I am only talking about efficiency and execution and not societal value (differing goals). I also recognize that there are many tasks/goals that require larger organizations and more longevity.