best way to batch test continuity?
A partial answer.
I have made test boards for shorts to VCC or GND for boards with multi-way connectors. I connected 4 adjacent signal pins together, to a toggle switch (one of a dozen, testing 48 signal pins), to a common rail I could connect (via a DMM set to buzz) to +V or 0V with 2 more switches.
Plug in a board, close all the switches, connect the common rail to each supply in turn, and you instantly know whether there are any shorts to GND or VCC.
If the buzzer sounds, switch off each switch in turn until it stops : that indicates (within 4 pins) where the short is. (Close all the other switches you opened in case there's more than one short!)
Having localised the fault to 4 pins, you're on your own...
But on a batch of 100 boards it paid for itself many times over.
Having eliminated shorts to power and GND, the board was safe to power up, and test in operation to find opens or signal pins shorted to each other, using a test program.