No. But, after reinstalling Chromium like four times (trying the latest normal installer, a source-forge auto-updater package, the 64-bit version, etc.), disabling all extensions, disabling all plugins (save the Unity Web Player), reinstalling the Unity Web Player, clearing the cache, deleting the whole User Data folder, restarting Windows, terminating the Unity Player process then refreshing the page, etc, it still failed (i.e. got stuck on the first frame, not loading the loading bar–as you described).
But then, I tried downloading and running the “Portable Version” of Chromium, and it worked! So, I guessed: it must have been something in the registry, or some hidden folder that Chromium used (and that it created anew for the portable version, in its own folder).
But then… oddly… when I launched that Portable Version Chromium as a normal/install-mode Chromium (that is, without using the “As Portable” launcher), the Unity Web Player still worked anyway! So oddly, whatever was wrong was not solved specifically by launching Chromium in portable mode… but rather, something must be different about the “Portable Version” package itself. (when launched in non-portable mode, it uses the same folder as a regular install; so yeah, it’s using the same folders, and yet doesn’t have the issue)
So then what was the low-level solution? I’m not sure. All I know is that for me, downloading the Portable Version of Chromium (as found here–the “Chromium Portable (64-bit) (Muhammer Ayes build)” one: http://chromium.woolyss.com), but then running it in normal/install mode (that is, without using the root-folder launcher) got the Unity Web Player to work for me.
So, I’m not sure if it would work the same for a Chrome build, or if they even have the same sort of Portable Version builds, but if it’s really bugging you, it might be worth trying something of that sort (using different Chrome builds/packages).