can .webloc file icon show a useful web page preview? Assuming Finder --> View --> View Options --> Show Icon Preview is enabled, is it possible for a .webloc file icon to show a preview of the referenced web-page?

I think the browser would have to do this, when I drag from the browser URL box to the desktop.  But so far, with multiple popular browsers, I see nothing but not-very-informative standard icons.  For example, the Safari compass.

Does any popular browser support this, say,  through a preference setting or an add-in?

Motivation:  I can now sort through a bunch of TextEdit files by putting them in a folder, view them as icons, and using the Finder option to increase the icon size until enough is visible -- document "shape" and larger text. Why not .weblocs?

Obviously, the browser-created .webloc file name should be useful, but often it is not, and --when I'm collecting a bunch of .webloc files for reference use-- it is a bit of a pain to have to edit the name of each.

Obviously the icon content would be a snapshot at the time of collecting the .webloc.  I assume there is no way to see the current content.  OK by me.  Thinking about the potential for just a minute, this possibility seems at minimum creepy ... to very undesirable, unless completely under user control.  Well, anyway, the kinds of reference pages I want to collect and sort through generally don't change much.

TIA
I did finally locate an application that provides the solution you are looking for.  The application isn't hands-free initially as it is a command line program.  

setWeblocThumb generates custom icon consisting of a preview of the web page that the webloc file points to.  It does not do this universally.  You'll need to 


point it at folders that you'd like it to watch
Assign a folder action to the folder that you'd like setWeblocThumb to watch and have it run setWeblocThumb.  


It's fairly straightforward, but is a bit more than simply downloading and installing an application.  The good news is that it's free.