Most reasoned arguments for government intervention happen on welfare-theoretic grounds, so why not use those types of "unseen" costs or benefits as reasons for government intervention? When a people pay for immunizations, what they see is the cost in both time and money to get the shot, while what stays "unseen" is the population of other people who are not going to contract the disease. When an oil rig is opened, what is seen is the growth in economic activity and jobs, while what stays unseen is the pollution that contaminates other towns. When Wall Street firms overleverage, what they see is the higher return when times are good, what remains unseen is the systemic risk that devastates the system when times are bad.
In these situations, government policy can allow the market to see what was previously unseen. Subsidized immunizations show parents that immunizing their children can save the lives of others. Carbon taxes show companies that their oil consumption hurts the health of others. Higher capital requirements show banks that their higher return can lead to the fragility of others. Often times, transaction costs are too high for the Coase theorem to internalize these externalities, and it becomes necessary for the government to try to reveal the true costs of actions to the market.
Of course, perhaps there's a government failure, perhaps the government does not have the necessary information to intervene. But at this point, Bastiat stops becoming a sufficient argument, therefore we need to look at the empirical data. For this reason, I read Bastiat and am left with "meh"; it is much too general to strike at truth.