The civil service is legally protected from the influence of democracy and the legislature has by and large ceded the real implementation of law to the civil servants themselves.

That's plainly not true, at least in EU public education, which I am most familiar with. Most official rules and protocols for professionals in public universities and schools are part of the law and not arbitrary decisions from a manager. These laws are chosen democratically and are revised relatively often, with major overhauls every ~5 years, for better or worse.

These institutions are also constantly dependent on grants and budgets that need to be approved by the elected government.

The lack of flexibility can be a bit oppressive at times, and there can be severe penalties for ignoring rules, even on small protocol lapses, since they are the law. But it's mostly fine in practice, it's not a significant bottleneck to efficiency.

The result is that public education is generally much higher quality than private education. Private ones just tend to be for students with grades that are not high enough to get in a competitive public programme, not that they are very competitive, there's plenty of room. The qualifications required from professors also tend to be much higher in public education than in private, and they get more room to breathe to focus on their specialized courses and research, whereas private professors are overworked and used in areas they are not qualified in.

And the difference between the top and bottom educational institutions is so negligible that top-performing students can happily stay local and be successful. Perhaps there is a difference in the network you might acquire, but not in the quality of education.

In Belgium art grants are basically determined by committees of peers organized by largely politically independent bureaucrats at the many different levels of government we have.

It's all navel gazing, entrenched interests and nepotism. Effectively the art sector is given tax money to perpetuate their own interests with limited democratic control.

Yes indeed, I was directly answering the parent comment, but the post is about "Who pays for the arts?" so you have a good point, I've read your other comments as well.

Generally subsidising art properly seems like such a fundamentally hard problem. I think a good policy would be to balance how much is given to popular art, fringe art and academic art, and have very different criteria to judge their merit.

I think they all have value, and I cannot see a better way of making decisions about "academic" art unless it is by their peers, however elitist that is. This can be counterbalanced by promoting some popular art, which is judged against its, well, broad popularity, a bit more democratic, but that has plenty of perverse incentives too. And what about "fringe" art? The truly innovative stuff. There is really no way to judge the merit of that by contemporaries, so perhaps we should prioritize helping people that can prove their commitment to an artsy lifestyle and have consistent output, whatever it is, as long it is not too derivative.

But think about what actually democratic art subsidising would be, having elected officials only in charge of it: effectively propaganda for the party that is currently in power.

There is a similar problem with funding science, which I believe is much more dangerous and has an enormous impact. You either have peer committees judging scientific merit by whatever criteria they feel is valid with no accountability, or you focus on performance metrics (citations, papers in reputable journals...), or again you have unqualified elected officials making rather arbitrary decisions based on public perception. They are all kind of terrible, it's such a hard problem. I guess the solution, again, is counter-balancing the terribleness of each option against the others, which is kind of what we are doing now. And it is terrible, but it kind of consistently works, quite inefficiently. But relative to what? Is there something better?

A free market of voluntary transactions solves this problem

Sure partially, that is working well too, but there are many things in arts and sciences that have long-term societal value and don't perform well in markets.

That's the whole reason why subsidies are a thing, and they are extremely effective for all kinds of common good. Indeed, that's the whole point of governments generally, to enforce the common good that falls through the cracks of market dynamics (and well, to ensure a fair market in the first place).