I finally figured out what else can cause this missing text problem.
The game stores it's fonts and localization information in "C:\Users\*User*\Documents\This War of Mine", instead of the game installation folder, where it would actually make sense.
On my machine with the missing text problem, both C:\Users\*User*\Documents\This War of Mine\LocalizationPacks and C:\Users\*User*\Documents\This War of Mine\LocalizationBinFonts were empty for some reason. And on my machine that was working they were filled with various fonts and data.
I deleted that folder from Documents and re-installed the game with Galaxy, which fixed it finally by re-installing the missing fonts and information to the Documents folder.
So basically if something happens to that folder in Documents, all in-game text will disappear. This is likely the reason why some people have to run the game as administrator, so they have proper access to the Documents folder where the fonts and localization data got installed.
I'm not sure why the devs are storing required game data in that location, it's horribly bad practice.
Thanks a lot!!! I was so sad that I could not play this great game anymore! This did the trick, well done.