The behaviour can be in line with desktop.. ie. click extension to get a choice to add current page to "favorites" (assuming extension manages the new tab as well, which is where the shortcut would go). Only when in the new tab page, I'd allow drag and drop of the icons.

It's pretty nutty to think just how crude the Google Chrome default tab page is. I'm not sure why it is that way. I don't think the page screenshot is that much helpful. Some sites look good and the thumb is pretty recognizable, others are relatively random.

Actually, I wonder if an extension can pull the "apple-touch-icon-precomposed" hi dpi icons in the meta tags?