Do you know what am I missing?
The AirPort Extreme cannot perform as a wireless to Ethernet bridge when it "joins" a wireless network because the Ethernet ports are not enabled in this type of configuration.
The Extreme will perform as a wireless to Ethernet bridge when the device is configured to "extend" a wireless network. The rub here is that the "extend" setting will only work if you have another Apple router providing the wireless signal.
Ironically, a less expensive AirPort Express can do what you want, with a "hidden" feature called ProxySTA, but you will have to set it up manually to "Allow Ethernet Clients" using AirPort Utility 5.6 for Lion.
I do not think...but have not verified....that the normal version of AirPort Utility 6.x packed in with Lion has this option.
If you need more Ethernet ports, you can connect a 5-port, 8-port etc. Ethernet switch to the Express.
Post back if you need more info about this option.