This is now appearing to be a Latch-up issue rather than ESD as initially suspected. The devices appear to fail with roughly the same symptom; the Vdd to GND pins of the device measures anywhere from 40 to 80-ohms, and high impedance to any inputs or outputs, or to the substrate pad. Some or all RESET outputs do not assert with sense input conditions correct for assertion.
I can now reproduce the failure by invoking soft power shutdown, then momentarily disconnecting and re-connecting the DC input power to the card. The chip VDD is the first rail that comes up straight from the input DC and normally monitors the various rails on the board. As it turns out, some of the sense inputs as well as output pullups are fed from rails that have lots of capacitance and do not discharge very quickly. Cutting the input DC and re-applying it soon after results in the chip briefly being exposed to pins being biased above the immediate Vdd level. Those bias voltages are well under the Vdd max of 7V and the chip should be able to tolerate this according to the TI answer on Nov 14, 2017 to a post regarding sense inputs being powered when Vdd=0V.