OK, I might have solved it. We'll see how it going to work in the future. Basically, when trying the data on another hard drive, I noticed that macOS was asking me to grant access to RStudio the new hard drive.
It got me thinking, from the suggestion you made, that there was a problem with the privacy or accessibility of the drive. So I went in System Settings -> Privacy & Security -> RStudio. Then under Removable Volumes I switched it off (Restarted RStudio) and then on again (Restarted RStudio). Then I had access to the files!
Hopefully this was the fix. Although it's not clear to me why it was doing this thing...