There are many methods of elucidating the value of public goods to people. It's certainly harder than for normal goods, but it's not a blind shot in the dark.
Yes, you can use "scientific" measures such as child mortality, which even the most hardcore subjective value proponents will agree is something that 99.99% of us value.
Minimizing child mortality by increasing spending on public goods is all well and good, but there comes a point at which each additional tax dollar will buy an increasingly smaller reduction in child mortality. The problem is that governments don't know where to stop and have a tendency to increase spending forever. Zero child mortality will never be achieved, but a lot of governments set these kind of aims, regardless of the cost. The mistake they are making is that they are only measuring what they choose to measure and ignoring unseen effects. Spending on public goods always comes at the cost of something else. Perhaps reducing child mortality has the result of increasing road deaths, because there is less money available for traffic lights.
These "scientific" measures only measure isolated sectors; there is no good scientific measure for the utility of the
entire economy.
Needless to say, child mortality would also be reduced in a free market.
Btw, the above is the best case. In practice, governments often consciously enact policies that are
scientifically proven to be unfavorable in terms of above measures because they are pandering to the caveman instincts of ignorant voters who have little personal stake in the public good.
A classic example is the blocking of cheap and highly effective needle exchange programs by right wing politicians, even though needle exchange is scientifically proven to reduce the chance of catching hepatitis and hiv,
even for non-drug users.
And I'm sure you wouldn't love the underproduction of public goods when you saw how fucking miserable your life would be with out any environmental regulation, public trash and sewer systems, utilities in some cases, fire and police protection, and public health.
Fucking people espouse all these ridiculous ideas from a featherbed.
These ridiculous ideas don't come from a featherbed but from my personal experiences.
I used to live in a country where I suffered from heavy air pollution, where the trash piled up in the streets, where the sewers were simply emptied into the sea, where I didn't feel safe in the street because the police didn't do their job, and where people died from treatable diseases because public hospitals had 10 years waiting time.
Despite the fact that people are forced to pay 1/3 of their income for those "services", as soon as they can afford to, they get private insurance, send their kids to private school, and move into gated communities that do a much better job of providing security and trash removal.
A free market doesn't do a perfect job of providing public goods but government does an even worse job.