(How) does LM386 allow an input past its negative rail?
Section 6.3 in the data sheet says that an acceptable input voltage is
+/- 0.4 V. That is just below a single diode drop, so it would confirm your theory.


Yes, that confirms what I thought about the PNP BJT input stage. It will sail close to BJT saturation with an instantaneous value at -0.4 volts but, it'll still work: -


As can be seen, the input signal falls below 0 volts quite significantly (-200 mV) without any problem or distortion in the output waveform (connected to an 8 Ω load). Signal is 1 kHz and transistor models are BC557 (PNP) and BC547 (NPN). Current source is 5 mA.

Simple answers are the best

Simple answers are usually the best but, not at the expense of truth.