When you say "generic" do you mean a blank, white icon? and not the one posted in my original post? If so, you do have a different problem than us in this thread, as this issue was introduced with Safari 14.0.1 and was fixed with Safari 14.0.3 update. The only suggestion I can make is OnyX for Catalina. It is a free app and I have used it for years with excellent results. Use at your own risk (don't you dare blame me if it bricks your rig!!) One of the Maintenance options is to rebuild the "Launch Services database", which per their help page states:
"You can Rebuild Launch Services to correct miscellaneous issues: duplicated display or appearance of deleted applications in the shortcut menu's Open With command, incorrect icons for certain files and folders, or opening of the wrong application upon double-clicking a document."
As with all 3rd party programs like this, definitely make a backup before use. Good luck!