Yes, same result as manually putting a gupdate.gcd into the Garmin folder, in a healthy device WebUpdater writes the new gupdate.gcd to the device's Garmin folder and the the GCD is initiated on next boot attempt to update the firmware (system software). That's happened with your 3xx/6xx so it means this device is probably functional to some degree. So far so good.

Now what about this question: "I assume Mass Storage Mode is no longer available either now?" It's important to know if MSM is available now with the original (not cure) firmware loaded.

Edit: I now see you've edited your post to add that you put a backup onto the device so I guess you must still have MSM available. I hadn't notice the edit before writing the above. So now you need to see if we can get this thing booting with 6xx fw started by 3xx boot. I'll make a hybrid SD kit for you to use, standby.