No, I do understand what you are saying; however, I disagree with it. In order to create wealth, you must create something that people will pay more for than it costs to manufacture or produce (whether it be a product or a service).
If a researcher in a government lab creates a type of genetically-engineered corn that will resist pests while reducing the amount of pesticides needed, then this would provide the country with wealth in the form of more plentiful and inexpensive food supplies.
Military research has led to many marketable inventions and technologies (i.e., thermal imaging, the internet). If these products produce more income for the country than was paid into their research, then, once again, wealth is being created.
Government-funded research can pay dividends to the country, just like investments in the stock market pay dividends to the stock holder. (Unfortunately, they tend to do it less efficiently than the private sector, but that is the topic for another discussion where out opinions would probably be more in line with each other

.)
Without teachers there would be significantly less professionals in the world that in turn create wealth. Just like a good business invests in important infrastructure because they realize it would be good for the bottom line, so does the government. By investing in teachers, we (as representatives of our respective nations) are providing a solid infrastructure for the private sector that will pay off down the road.
While the private sector is vital for injecting money into the economy, the public sector is also a vital support for the private sector. They are in a symbiotic relationship; without each other, neither would flourish.