Piece of cake, I thought to myself, disregarding the fact that most of the time the pieces of cake are tough to chew when it comes to SharePoint. I created some site columns and added them to the list, positioning them next to the original ones, with the intention to simply open the list in quick edit mode and copy values from list columns to site columns. Easy enough!
However, the list being a custom list originating in SP2010, the quick view was unavailable. I contemplated on exporting the data to Excel, modifying it there and creating a new list using that sheet. Before attempting that, I decided to google just a bit in hopes of finding a solution. And I did, hurray for Saludeen Rajack!
Since I had already tried most of the tricks in the blog post, in vain, I crossed my fingers and opened the list page for editing, opened list web properties and added clienttemplates.js to the JS Link property of the web part (at the very bottom of the Miscellaneous section).
And lo and behold! First of all, the list search box appeared at the top of the list and secondly, the quick edit button became active.
Now I could proceed with my plans properly.