The problem has been 'solved' only to the extent that people accept 2nd-year Public Finance as it's taught to 3rd years economics students (which stops short of two of the most critical problems with the theory - bureaucratic capture/corruption, and war - and a third... the general equilibrium effects of a very large budget-insensitive actor in goods and factor markets).

The basic Pub Fi model is that the existence of publicness characteristics in some goods means that some markets (health, education, defence) are underexpanded, and some (pollution, nuisance) are over-expanded, due to social benefits and costs not being taken into account... which - on a purely utilitarian basis - are to be resolved by .gov... from there you learn that diminishing marginal utility of money means that it imposes the lowest "excess burden" on society if you tax income progressively. The nyou add up all the little Harberger (welfare) triangles and declare that the State serves a utility-optimising function.

HUGE problem (which I would have thought any rational individual would have spotted) is that the moment you introduce a '.gov', you set on the table a giant pot of money and power. In democracies you then say "OK, this pot of money and power is open for competition: all you have to do is convince 30% of the voting public, which is roughly the proportion of the population who can't read the instructions on a tin of beans (read any LISS/ALSS survey)... and if you lie your head off in doing so, no biggie because nobody will punish you for it."

Who gets attracted by that set of incentives? Sociopaths... and they're the wrong people to have in charge of the instrument that decides which things "are of utility to everyone" (and more to the point, decides how much of these 'things' will be produced... or more accurately how much will be spent on them).

Others' education is not of utility to me, beyond basic literacy (which the State is really bad at teaching, especially if you look at value for money) - and yet States operate high schools and universities, the value of which is entirely captured by higher lifetime incomes (i.e., it's a private benefit).

Others' health is not of utility to me, except (perhaps) for downstream effects from vaccination and some minimal level of acute care- and yet States run hospitals with cardiac units (again, things with solely private benefits).

And State-furnished money is specifically and deliberately of lower utility year on year (it is not a store of constant value).

You seem to think that in the absence of a State, the things you mention will 'go to zero' - that's not what "public goods" implies. It simply implies that the amount produced of those goods will be lower than a perfectly informed agent (with no power in or effect on factor markets) would choose - that is, that the level of production that actually occurs will be sub-optimal, not that it will be zero. And always in utils - i.e., based on the assumption that taking a util from a sickly child and giving it to Warren Buffet, is net-social-utility-neutral.

Also... what do States do intertemporally? Due to the tendency of bureaucracies to grow, States always and everywhere grow outside of the bounds of their "defensible" spheres of action. Taxes rise, output quality falls (as usual for coercive monopolies), debts accumulate. Cronies are enriched.

And then there's war: modern, industrial scale, baby-killing. All of those Harberger triangles that were accumulated by 'optimally' expanding the public goods are blown to smithereens by wasting them on cruise missiles and travelling bands of State sociopaths.

There is a vast literature on the optimality of furnishing all 'critical' State functions by competitive processes - courts, defence, policing etc - Rothbard, Hoppe and others in that space make it abundantly clear that the State is a net negative, even if you use the (ludicrously simplistic) utilitarian framework to analyse it.

So yeah... the problem has been solved: just not in the way you think it has.