Yeah, the BIOS sees the drive.
I found another repository of HP diagnostics tools in some recovery media my wife must have made for the machine when it was new (its CD-RW hand-labelled "1 of 3", but only have the first one, so....). In any case, it does have some tools, which DO see the HD there and which are able to run tests on it (and memory and other components if I so choose).
SMART: SMART Status Test: Passed
SMART: SMART Short Self Test: Passed
Hard Drive Test: Random Seek Test: Passed
Hard Drive Test: Funnel Seek Test: Passed
Hard Drive Test: Surface Scan Test: Passed
Hard Drive Test: Surface Scan Test: Passed (yes, there appear to be two of this last test with the same name)
FWIW, this particular set of diagnostics thinks the HD is just fine, but if I let the software "Restore to factory defaults", it craps out pretty soon:
"Error Ox400110020000100A"