Changing download method from Private to Public: Image module and screwy paths

This is a long-running question I've had. I can't find a forum post with exactly the same question, though this one comes close: http://drupal.org/node/489850

I have a site that's used the Private download method from the start. I'd like to switch to Public, so I can enable CSS optimization. Easy enough to do, and most images – logo, and images inserted into text via TinyMCE – are unaffected.

But I also have many nodes with images attached via the Images module, which adds an "Attached images" upload option to nodes. (Side note: Why it's labeled "Attached images" when it appears to allow attachment of only one image, I don't know.) Paths to those images break when switching download methods (which, fairly enough, warns in advance of such trouble).

The site is one of several on a single Drupal installation, so there's a path

drupal_directory/sites/site.com/

The file system path is set to

sites/site.com/files

The Image module adds to the above path – adding "/images" in my case – resulting in

sites/site.com/files/images

Images so far have been uploaded, by whatever means, to

drupal_directory/sites/site.com/files/images

This results in images which appear on pages to have the underlying URL

http://www.site.com/system/files/images/image.png

(It's my understanding that the "system/" that gets tossed in there is normal Drupal behavior.)

So far so good. But when switching to Public download, images uploaded via the Images module cease to display; the placeholder "broken image" square that appears instead has the wacky URL

http://www.site.com/sites/site.com/files/images/image.png

Even uploading a fresh image via "Attach images" doesn't fix things; the newly uploaded image still gains an unworkable URL. I'm stumped as to how to tell the Image module to fix that URL and look for the images in the normal place. Anyone know a solution?