I've just found out what the problem is.
The 3G stick stops working after a while because of a mechanical problem inside the tablet. It is a major design fault which may cause any Archos G9 (both 80 and 101) fail to provide connectivity to the back USB host port. The reason? A poorly chosen connection ribbon between the mobile USB connector and the PCB.
Just look at these pictures.
First, the layout. You can see the ribbon about the center of the image, with the USB connector on the right side. Keep in mind that the USB connector slides back and forth in the USB stick area, by design. That is, the ribbon will be continually moved, which places a certain mechanical tension on it.
The following is the ribbon on its back side, featuring the data and power lines for the USB. Note the very thin lines in the middle. They are the data lines for the USB port, while the other two are the VCC and ground connectors. The very thin lines are very easy to break.
And this is how the data lines became broken after a while (less than a month of 3G usage in my case), because the ribbon is moving at each insertion and removal of the USB stick. Notice the several interruptions in the middle lines. The outer lines (VCC and GND) are not yet fully interrupted. The effect: the 3G stick will be powered, but it will be unable to communicate with the tablet. This is the exact behavior I noticed on my tablet.
ARCHOS, please fix this thing. AND send me a replacement cable/ribbon. It's a shame, you should have used a real cable between the connectors, instead of that frail ribbon.
I hope this post helps other users to understand what's not working with their tablet. This is obviously a hardware fault due to improper design, and obviously cannot be fixed by software
So, if your tablet exhibits this problem, send it for repair.
I believe this to be a generalized problem. As you can see from the excellent teardown by Harfainx for the 101 G9 (viewtopic.php?f=85&t=61635&p=396078&hilit=teardown#p386593
), the ribbon is the same in that tablet, too.
AGAIN, Archos, I hope you will send me a new compatible connector to fix the tablet myself...