Amazing! This turned out to be the answer. I installed the ASAPI driver from the URL you listed, switched AudioGrabber to use the "ASPI Manager" instead of "Windows calls", selected "buffered burst copy", and everything now works great. (I already had "spin up disc" turned on.)

It even successfully ripped a used CD I have that my old computer couldn't even begin to rip properly.

Paging Mr. Fabris: please put this in the FAQ somewhere!