Digital signal appears to improve with length of wire - troubleshooting multiplexed LED display
I attempted to replicate what the voltmeter was doing to the signal based upon the help received here. I tried connecting the signal pin with ground via 1M resistor, then 10M ohm, then 10M ohm plus a capacitor (which should've been approximating the voltmeter's effect), but none worked.

Removing the resistors and placing the 100nF capacitor as close as possible between the two signals did the trick. Strangely, however, holding the cap with my fingers and pressing it on the connections worked, but after soldering it on it didn't. Tried it three times to no avail. The difference in connection location was just millimeters, so it was very strange. However one difference between the manual testing and soldering the cap onto the board was the length of wires on the cap. Leaving it full length and soldering it on works. So seems like those extra couple of cm of wire and a well-placed cap was what was missing!

Although I've understood that in general one shouldn't be placing caps on signal lines, so that's why I didn't try this earlier. Weird that it works, but I'm very glad!