If it’s free then you’re the product …

That's a interesting way to say creating a healthy contributing member of society that leads to future gdp.

How would this apply to, say, public libraries?

Yes, in the case of a business giving out free services or things. But, government is not and has never been a business so this doesn't apply in this case.

That's true of private enterprise that has a profit motive. The "product" here is healthy well-adjusted citizens that can one day be workers. The public sector is not the same.

Although, you could also say the "product" are additional parents that can work.

This but unironically

Really? You're the product of, say, your local fire department?