A bit of sleuthing with an oscilloscope led to an unusual conclusion. My first problem was an oscillation on the voltage regulator. I used an LM7805 regulator, but raised the ground pin up with four 1N4001 diodes to give about 7.5 volts. On the voltage input pin, I could see a .5 volt peak-to-peak oscillation at about 2 MHz. There was a smaller variation on the output pin. I pulled a couple of .1 uF multi-layer ceramic caps out of the junk box and proceeded to put them across the input and output to ground. No more oscillation in the audio amplifier.
While I had the rig on the bench, I was also trying to diagnose some harshness in the presence of strong signals. What I found was that the AGC circuit was allowing some of the received audio to make it into the AGC input of the MC1350. Another .1 uF cap on the AGC input of the MC1350 helped to smooth that out.
Another issue I tackled was the BFO frequency trim only allowed a center frequency of about 850 Hz. I added a 10 pF capacitor across the adjustment trimmer and could tune down to about 700 Hz, which is much more listenable.
Only issue I haven't figured out is a third-harmonic response around 2100 Hz in the receiver passband. Probably need another capacitor somewhere else.
In any case, about an hour of work and the receiver works well with the internal speaker, and is much more listenable on the phones.