The obvious culprits would be firewall or an anti-virus, but adding Discord to exceptions list or even disabling them doesn't fix the issue. More specifically, it shows a progress bar and doesn't give any errors, but the folder with overlay files is missing. So the way it should work is that you activate the overlay through the app itself, and it downloads all the necessary files.
However, while trying to enable overlay something weird happens. I have no problems with installing it and I encounter no issues when it operates.
The app in question is Discord, a VOIP software.