But of course that only tests the output lines of the cables, and your problem is in the input. I would use an ohmmeter to check the connections from the input phone plug to the DIN connector.
But considering that you've tried two different cables, it does sound like the problem may be inside the Model III, which will be harder to diagnose and fix. You can find the schematics and theory in the
technical reference manual.
The 1500-baud and 500-baud load routines rely on different components on the motherboard, so it's possible 500-baud is broken but 1500-baud isn't, so try both. Also, 1500-baud is less volume-sensitive and generally more reliable, so I'd try to get that speed working first. (The 500-baud and 1500-baud output, and sound effects, are all using the same output components, which are separate from and simpler than the input circuit, so the fact output seems to be working doesn't really tell you anything about whether your input is working.)
And as trs-80.com said above, you may have better luck testing the TRS-80's input by using clean audio output from a modern computer, using something like the Playcas utility. That would eliminate the possibilities that the problem is that the tape recorder's fidelity isn't quite good enough, or that the recording is bad because the TRS-80's output was slightly garbled.