Healthcare and education are not only important; they are vital. Most of us would prefer to live in a society that so organized itself that these services were accessible to all its citizens. This is not, however, the same as saying that they should be produced and delivered by the state.
When the state goes into the mass production of services it tends to put them into the political domain, where they can be influenced by ideological or interest groups. Politicians can manipulate them to secure electoral advantage. They can be effectively captured by producer groups such as teachers' or healthcare workers' unions, to the detriment the citizens who consume them and the taxpayers who fund them.
When the state does mass-produce services, they tend to be standardized. It is easier to have a one-size-fits-all output than one that caters for individual preferences and allows a variety of choices. The private sector, by contrast, tends to find different niches being filled by a variety of producers, allowing consumers to choose the level and quality that suits them.
The state can fund education without producing it by giving people vouchers to cover the education of their child, or by routing the funding to the school of their choice, as is done in Sweden. This leaves the schools independent and in control of the education they offer. Healthcare can similarly be financed through insurance or refunds, without the state having to own hospitals and employ nurses. Again, countries that do this tend to have more variety and choice.
Education in state-run comprehensive schools is not very good. There are some good ones, but a great number that fail their parents and children by leaving them ill-equipped for life. Healthcare in state-run hospitals varies hugely in quality, with recurrent exposés of inadequate care or neglect.
Funding for state-produced schooling and healthcare depends on what politicians think taxpayers will tolerate. Their output does not depend on what customers want. Far from being too important to be privately provided, healthcare and education are too important to be publicly provided.