Why does the government produce public goods ?

Because the market economy doesn't meet all our needs. When people need something, and the market doesn't provide it all, or it does provide it but it's out of their reach, they look to government. Every modern, developed, successful country has some mixture of capitalism and socialism, each in its own place, to do what each does best.

Here in the US, for instance, we are probably the most capitalistic country in the world. But we have 'socialized' education, roads, police and fire protection, air traffic control, electrical and water projects, epidemic prevention, etc.etc. Because over the years we've found that this is just the best way to provide these services to the people who need them. These are generally things that we want everyone to have access to, regardless of their ability to pay.

And capitalism, as we know it, couldn't exist without a considerable amount of help from government. Much of our R&D is paid for by the taxpayers. Our transportation infrastructure--railroads, airlines, canals, harbors, interstate highways, light rail systems, etc. could not have been built entirely by private investment. But they were needed, -everyone- benefited from them, so everyone paid for them. Kids need an education even if their parents can't afford a private school, so we built a public school system. etc.etc. A pure 100% capitalist economy simply won't work.