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.
(This is a semi-serious introduction to the metaethics sequence. You may find it useful, but don't take it too seriously.)
Meditate on this: A wizard has turned you into a whale. Is this awesome?
"Maybe? I guess it would be pretty cool to be a whale for a day. But only if I can turn back, and if I stay human inside and so on. Also, that's not a whale.
"Actually, a whale seems kind of specific, and I'd be suprised if that was the best thing the wizard can do. Can I have something else? Eternal happiness maybe?"
Meditate on this: A wizard has turned you into orgasmium, doomed to spend the rest of eternity experiencing pure happiness. Is this awesome?
...
"Kindof... That's pretty lame actually. On second thought I'd rather be the whale; at least that way I could explore the ocean for a while.
"Let's try again. Wizard: maximize awesomeness."
Meditate on this: A wizard has turned himself into a superintelligent god, and is squeezing as much awesomeness out of the universe as it could possibly support. This may include whales and starships and parties and jupiter brains and friendship, but only if they are awesome enough. Is this awesome?
...
"Well, yes, that is awesome."
What we just did there is called Applied Ethics. Applied ethics is about what is awesome and what is not. Parties with all your friends inside superintelligent starship-whales are awesome. ~666 children dying of hunger every hour is not.
(There is also normative ethics, which is about how to decide if something is awesome, and metaethics, which is about something or other that I can't quite figure out. I'll tell you right now that those terms are not on the exam.)
"Wait a minute!" you cry, "What is this awesomeness stuff? I thought ethics was about what is good and right."
I'm glad you asked. I think "awesomeness" is what we should be talking about when we talk about morality. Why do I think this?
"Awesome" is not a philosophical landmine. If someone encounters the word "right", all sorts of bad philosophy and connotations send them spinning off into the void. "Awesome", on the other hand, has no philosophical respectability, hence no philosophical baggage.
"Awesome" is vague enough to capture all your moral intuition by the well-known mechanisms behind fake utility functions, and meaningless enough that this is no problem. If you think "happiness" is the stuff, you might get confused and try to maximize actual happiness. If you think awesomeness is the stuff, it is much harder to screw it up.
If you do manage to actually implement "awesomeness" as a maximization criteria, the results will be actually good. That is, "awesome" already refers to the same things "good" is supposed to refer to.
"Awesome" does not refer to anything else. You think you can just redefine words, but you can't, and this causes all sorts of trouble for people who overload "happiness", "utility", etc.
You already know that you know how to compute "Awesomeness", and it doesn't feel like it has a mysterious essence that you need to study to discover. Instead it brings to mind concrete things like starship-whale math-parties and not-starving children, which is what we want anyways. You are already enabled to take joy in the merely awesome.
"Awesome" is implicitly consequentialist. "Is this awesome?" engages you to think of the value of a possible world, as opposed to "Is this right?" which engages to to think of virtues and rules. (Those things can be awesome sometimes, though.)
I find that the above is true about me, and is nearly all I need to know about morality. It handily inoculates against the usual confusions, and sets me in the right direction to make my life and the world more awesome. It may work for you too.
I would append the additional facts that if you wrote it out, the dynamic procedure to compute awesomeness would be hellishly complex, and that right now, it is only implicitly encoded in human brains, and no where else. Also, if the great procedure to compute awesomeness is not preserved, the future will not be awesome. Period.
Also, it's important to note that what you think of as awesome can be changed by considering things from different angles and being exposed to different arguments. That is, the procedure to compute awesomeness is dynamic and created already in motion.
If we still insist on being confused, or if we're just curious, or if we need to actually build a wizard to turn the universe into an awesome place (though we can leave that to the experts), then we can see the metaethics sequence for the full argument, details, and finer points. I think the best post (and the one to read if only one) is joy in the merely good.