New thought: Could this issue be PSU related? I am using an IBM XT power supply to feed the motherboard. The pinout for the P8+P9 connectors are identical, except for the fact that while the XT P8 connector has a KEY pin after the POWER GOOD one, an AT P8 provides +5v on the corresponding pin. As all +5v pins are connected together on the motherboard (if I have understood things correctly), this just means that an AT PSU provides extra current to +5 voltage.
As the 386 motherboard, small as it is and with much fewer discrete components than, for example, the 5160 motherboard (and since I lack power-hungry peripherals like MFM hard drives, et cetera), I thought that this should work fine, but perhaps this is what creates the A20 issue?