When I Get Info on a blank white WEBLOC icon on my desktop, I see blank white icons in the upper left and in Preview. This has been the case since I "upgraded" to Catalina from Mojave over a year ago and I've been working to solve this problem ever since (what a lousy way to spend spare time.)
Since I have people trying to help me now, I have a new piece of information that I just uncovered, which may change the complexion of everything:
I am the sole user of my computer. I introduced an additional fake user and restarted under that name. Lo and behold, any WEBLOC icons show as they should and are functional. When I return to myself as user, nothing has changed. This suggests to me that there is something wrong in my profile that is leading to the icon problem. Now, I'm looking for new ideas.
Yes, if it does not happen in a TEST user account then it usually means that the problem lies within your User/Library (aka ~/Library).
I had some issues with Safari related Finder freezing a few months back and deleting the Finder prefs file did the trick. Here is the Terminal command: (first locate the file to make sure it is there)
defaults delete ~/Library/Preferences/com.apple.finder.plist
make sure it disappears, then do a restart (or relaunch the Finder through Force Quit). You will lose a few finder settings. You should be able to also drag this file to the trash, but I have had bad luck with that for library files, so I use the Terminal to delete. After this, check to make sure the file was recreated in that location.
To be ultra safe, I would find this file under ~/Library/Preferences/com.apple.finder.plist and drag a copy to your desktop in case this fouls you up, but it shouldn't. You used to be able to delete the contents of the whole user prefs folder, but now it is not advisable.
Here are a couple links you can try, one describes the method above without the Terminal command.
https://macreports.com/mac-displaying-wrong-generic-document-icons-for-files-folders-fix/
https://www.jgoode.com/easy-fix-for-mac-finder-preview-thumbnails-not-displaying/
Good luck and make sure you backup before you try any of this.