You are starting with the assumption that government strives as hard to keep costs down, as private businesses do. That is not the case.
People who don't understand Capitalism just see that profit as an unnecessary cost, but it's the MOST important cost in the whole equation. The desire to make the profit, is what makes businesses compete harder on pricing. The desire to make the profit, is what drives the businesses to make products that are better than their counter parts. The desire to make the profit, is what drives the business to listen to their customers, and cater their products to them. In addition, the profit margins also provide competition of pricing. If you can't get your costs as low as your competitor, due to whatever advantage the competitor might have, you can still compete by simply taking less profit. The opposite side of that coin, is that businesses have to compete with profits to attract investors. If one company is more profitable, they will get more capital investment.
When the market is truly free, devoid of cronyism, it is by far and away the best way to get the best, cheapest products in to the hands of the consumers. The nations that did not believe in what I just said, are nearly all extinct. Reality > Feelings.
Does this mean we should privatize all industries? Of course not. Many lack the viable diversity of competition for Capitalism to work. Examples of this would be road maintenance, fire departments, health care, and others. These industries fail to have a model that makes private delivery a viable solution.
But, if you are saying we would be better off with one government ran automaker, for example, I present to you the Soviet manufacturer, Lada.