Wow, that's the most oversimplified attempt at discussing economics I've seen for a long time. Hint: economics is complicated - if it were easy we'd have a simple working system that everyone could agree one (or, at least, the people who worked out out would get very rich by betting on the economy 100% correctly and other people would notice) - and any explanation that simple is likely to be wrong.
Consider the following counter example:
The government builds and maintains a road between two places. This employs people, taking them out of the labour pool where they could be doing other things, so it's a cost. But it also allows the two places to trade more cheaply, which increases wealth production. Similarly, employers at either end of the road would have access to a wider pool of employees and potential employees to a wider pool of employers and so people would end up in more productive employment.
Now, would the same apply if private industry built the road? This is where it starts to get more complicated. First, who would build the road? It might be some consortium of businesses at both ends who wanted to use it. If so, then they might charge money to anyone not part of the consortium to use it, which would give them a competitive advantage, but be less healthy for the economy as a whole by producing a barrier to competition (and, most specifically, a barrier to entry for new companies).
It might be a third party that thought that the road would be profitable, who would run it as a 'common carrier' toll road. This, however, would provide a disincentive for people to use it. If they priced it too low, then they'd go out of business (which would discourage future road-building companies). If they priced it too high, then they'd make it unprofitable for some users to use it, however given that the cost of the road is now a sunk cost the economy as a whole benefits if as many people as would gain any benefit at all from it use it.
In some cases, the benefit to the economy may be significantly lower than the cost of the road, so it would not make sense for the government to make the investment. It's often difficult to make that call, however. In the UK, be Beeching Report identified a large number of unprofitable railway lines and, to save taxpayer money, the nationalised railway service closed them. Unfortunately, it turned out that a lot of the unprofitable lines were ones that got people from near where they lived to a more profitable line. When they were closed, people at the edges ended up having to buy cars, which meant that they no longer used the larger lines either, and so pushed those into unprofitability (and so there was a second Beeching Report some years later which repeated the entire mistake). The cost to the economy of no longer having a widespread, cheap, railway network is widely agreed by economists to be significantly greater than the savings from closing the lines.
A nationally owned private rail operator may have seen further ahead, but most likely they'd have had shareholders making the same demands: sell off the unprofitable lines and concentrate on the profitable ones. A larger number of smaller railway operators would have had similar problems, with the ones operating the unprofitable lines going out of business and reducing demand on the profitable ones.
