It is true that Blockchain technology has been thought as a replacement of a trusted third party like a notary or central register, but usually those are public registries: household ownerships, public offers... And those registries are usually exceptions or are treated in a different way (and usually they are explicitly regulated by each country). A few of them are even public, like the defaulters list in Spain.
Also, it is quite naive to think that an European citizen can erase "any personal data", that could be quite convenient to erase your fresh new mortgage or obligations. You are only allowed to ask for irrelevant information to be removed not for specific business related log even if they have your personal data in them.