> Can modern car radios go bad in a manner that the quality of
> the sound deteriorates at high volume
Certainly possible. But one of the first things I would check is that the
amplifier is getting a good solid 12v even at high power output.
For me, there would also be a distinct possibility that I'm trying to listen
at a higher volume than before. With the setup I had in my TR3A, there was
a fairly fine line between "couldn't hear" and distortion. And I know my
hearing has fallen off even further since then.
Unfortunately, it's quite possible that your new speakers can't handle the
power, as well.
For testing, all I can suggest is swapping with known good components,
unless you have access to a oscilloscope. I've got an older HP scope that I
picked up on eBay for $30. Only good to 20 MHz, but that's plenty for
working on audio.