Quite agree, I think that in many careers the university is not absolutely necessary, and in many others it is even an obstacle, although education is always necessary.
If you are in doubt about careers as delicate as medicine, you can see the historical example of the Kowloon Walled City, the small exclave of China in British Hong Kong, a territory that had no connection at all with the other side from China, so there was no government, no police, no law. Many doctors, medics, dentists, etc, operated there for decades without accreditation or degree.
Obviously this is not the example of perfection and not the way to follow, but it shows us how they could advance even in such delicate fields.
Education is always, absolutely always, important, but that does not mean that the only way to obtain it is through an institution.
But as a practical matter, as things stand today, it is an absolute necessity to get a degree if you want to practice medicine and I don't think a better training mechanism exists today for that purpose (which isn't to say that it is ideal).
I am quite pragmatic. I don't think that many people today would confide in a doctor without university endorsement, and neither would I in a stranger. The current system is based on distrusting individuals and trusting the institution, so it would be impossible.
But I am convinced that education is much more necessary than a university degree, and at some point not too far away, we will be able to do without them. And this does not mean eliminating universities, but on the contrary, giving them a merely educational meaning, in which no degree gives more opportunities to people, but rather education does.
I quite agree that institutionalized education is not always a solution even for advanced areas of study.
Kowloon Walled City is a very peculiar thing, and I think it warrants more study as a proof that anarchism can provide services we consider government services today. Its rampant crime was doubtless in large part a consequence of black markets created by government prohibition rather than proof that absence of government results in chaos.
I think anarchism, at least in any pure sense, is like any other utopian ideal. They tend to be more dystopian in practice. I could be wrong but I don't think anarchism scales well and it is vulnerable to non-anarchist societies in a number of ways.
Anarchism isn't a Utopian ideal, but rather the recognition that society functions in spite of government rather than because of government. Government claims responsibility for society's successes while blaming liberty for governmental failures. After all, government is just a group of people who claim a territorial monopoly in violence, and monopolies invariably promote waste and abuse. The record of wars, police states, arbitrary prohibitions, confiscatory taxation, failed centrally-planned economies, and other disasters of recorded history all fall at the feet of governments, not liberty.
I think wanting to implement a society with 0 government is a utopian ideal. I have no doubt society would function without a government but it doesn't mean most people would be better off. I agree with everything bad you say about government but it's easy to lay the blame for everything at the feet of government because in all of recorded history government has always been there (with a few very limited exceptions). However, the problem is with people and those problems won't go away just because you take away government. They will just take a different form. There will still be groups of people claiming territorial monopoly in violence or at least there's no reason to believe there won't be. I am not convinced that quality of life would generally be better if government just went away. Government has generally been reformed for the better throughout the ages. There's no particular reason to think that can't continue though it has certainly been a while.
Government has way too much power and I am all for reducing it at every possible turn. I just don't think the complete absence of government in the long run would ultimately be better.