Use the Original Driver CD
While this solution still makes absolutely no sense to me, it ended up being the only thing that worked. I had to use the original driver disc that came with my Linksys Wireless G USB adapter.
To install the drivers with the original CD, simply place the CD in your computer's drive and let the setup run to install the software, or just manually update the drivers using the one on the disc. This will allow your PC to connect to a wireless network so you can get other updates.
At the time this thing was made, Windows 7 hadn’t even been invented yet, but the drivers on the included CD actually worked with Windows 7 whereas the drivers Linksys said were made specifically for Windows 7 did not work at all. After those were installed, Windows 7 update found a newer driver and upgraded to that, and the wireless network connectivity on my desktop has worked fine ever since.
If this still doesn't work or you don't have access to the original driver CD, you may consider getting a PCI wireless card for your desktop.
(Image credit: Linksys Installer Screenshot from Author's PC)