For fifty years economists have emphasised that, because information is often a public good, that it will be undersupplied by markets. Hence there is a case for public funding and public provision. The first quoted item here is put in quotes by you while the second is a statement by you. I would be interested to know where you got these ideas from. I have never read or heard either type of remark from an economist.
Prices are only one way of achieving an allocation of resources.
Markets can fail and non-market mechanisms can then be useful. I do not know any economist who disagrees with this proposition. In the case of education we know that there are public good aspects and external benefits from having educated people. I will publish your correction, if I may. But first I would like to get clearer about a few matters and to answer your questions.