I had a bit of a derp and didn't wait the whole video through before posting a comment so I'll copy-pasta it here now:
Silly passing thought looking at your schematic and routing limitations and I thought I'd toss in my two cents:
Ideally it looks like you could knock down your test connections to 5 points and with a little clever cover make it not destroy the top surface continuity.
Mandatory traces on the bottom side for input positive, output positive, battery positive, and battery negative leaving no trace pairs along the ear to short out.
The trick for I/O negative; since they are tied together and looks like they are your flood plane on the top side would be to flood it "out" across all 10 units and viola! You can test all ten simultaneously after all. Of course this might be an issue when it comes to the routing since it might leave a nasty copper edge. I have no idea if your manufacturer can run a 90 degree bevel for the routing cut to leave a cleaner edge if you flood the whole top surface.
Final connection could be accomplished with a 24 pin (20 I/O positive pairs, 1 I/O common negative, 2 battery +/-, 1 NC key pin) IDC header since you obviously don't want to parallel all those sense inputs or device outputs.
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