Why VMware Player does not recognize USB devices?
OK, the problem seems to be a collision of the VMware USB Arbiter service with the AMD USB Filter Driver. I used the method described in section 5 to remove the filter driver using the AMD Catalyst uninstall wizard.
After removing the filter and restarting everything, a USB drive is now recognized and mounted by VM. I was able to play a MP3 file from the drive.
Note that as of now, I have not yet tested for any side-effects of the USB operation on the Win 7 system. Per the reference, there might be problems using USB devices after removing the filter.