That is not the case at all, though the fear of societal breakdown is the primary one that keeps people in a relationship of dependence with the state. This is precisely what those who run the state want and encourage: "without us, you'd be f**ked, and you know it." The starting point of libertarian thought is calling the state's bluff, or at least suspecting that they and their apologists might not be telling the whole truth.
It boils down to this: if you believe people are fundamentally so evil that they will not voluntarily look out for each other in times of strife or hardship, then there is nothing the state can do to change this reality, for the state itself is made up of these same evil people. If, on the other hand, you believe that the state in fact does a good job of helping the less fortunate, recalling that those who run the state are fundamentally no different to the rest of us, what reason do you have to suspect that ordinary people won't pick up the slack if called on to do so? The same point applies to education: on what basis do you make the claim that ordinary people will not see to it that education is made available to all? How do you reasonably claim that only those people who constitute the state care about this issue, and thus must be given the right to violate the private property of ordinary people in order to provide it?
Really, you take for granted that people will not care enough about others to provide affordable healthcare and education independently of the state. I challenge that this is a valid assumption, but contend rather that it is a dubious claim that needs to be justified.
A point that you perhaps fail to realise is that provision of services such as education and healthcare can only happen when there is an economic surplus, by which I mean excess production and generation of wealth beyond that which is necessary merely to survive. Looking specifically at education, part of this economic surplus is a level of general wealth such that people have time to spare beyond the tasks necessary to survive, time that some people can devote to teaching, and time that other people can devote to being taught.
Historically, up until the time of the industrial revolution, the economic surplus, even in the richest parts of the world, was not large enough to provide for mass education. To the great detriment of everyone over the last 150+ years, once there was an economic surplus sufficient to allow education of the masses, the state quickly assumed to itself the responsibility to provide it, with the result being a system which is expensive, inefficient, and that has changed little in a century and a half. It is also a system that fails many, many people, by assuming one size fits all, that everyone needs to learn the precise same content, that all children of the same age are at the same developmental level, and by almost completely ignoring the wishes and interests of the individual. The government (or the education bureaucracy) knows what you need to learn and when, and no correspondence will be entered into, aside from the constant political contest by various interest groups to be the ones who write the curriculum.
Private education, as it presently exists, is no solution. Government regulation of who can teach, what can be taught, how it can be taught, and who it must be taught to, prevents genuine innovation in the provision of education, innovation which is the norm in other sectors of the economy. Likewise, because we are all so accustomed to the way things have always been done, very few people are capable of imagining how it could ever be done differently, and done better.
