Finally cracked my fake nixie clock - the LEDs were never addressable at all

I went through almost the same thing with a cheap round matrix clock I picked up a couple years back. Mine had plain red LEDs behind a diffuser, driven off a shift register, and it took me ages to figure out because the traces ran under the display panel where I couldn't probe them. What finally cracked it was putting the meter in continuity mode and touching one probe to each LED leg, then hunting around the back of the board for the other end. Found three resistors feeding it from a driver chip that had zero documentation online, so I just mapped the pins by hand on a scrap of paper.

The display part sounds familiar too. Mine kept wrapping text halfway down the screen til I realized the library was set up for a 32x8 panel when the thing was relly 32x16 with two chained sections. One line of config and it all snapped into place. Since then I've flashed it a dozen times with diffrent layouts, and the wife likes the dimmed night mode better than the original ever was. Only tip I'd add is to photograph both sides of the board before you start desoldering anything, because I lifted a pad once and had to jumper it with wire.

I did something similar with one of those little round wifi clocks from a couple of Christmases back, but I skipped the tracing almost entirely. The brain on mine turned out to be an ESP8266 module right on the board, so instead of writing my own driver from scratch I pulled the flash chip with a clip and dug through the dump. That told me which pins the factory firmware was wiggling before I ever touched a probe. Took an evening or two less than your multimeter route, though honestly yours probably left me with a better feel for the board than I ever got. On the LED side, mine were also dumb common anode parts, and the one gotcha that bit me was brightness. Driving a colour pin straight low gave full blast red that washed out the whole diffuser at night. I ended up doing the dimming in software with a quick on and off pattern rather than touching any hardware, since the resistors were already sized and I did not want to desolder anything on a cramped board. Your screen offset thing sounds like the same class of mistake I made with a little SPI display on a weather project. Wrong height in the config and everything shifted half a screen over. Always the boring settings that eat the evenings.

Nice result on the tracing. One thing I'd flag from my own clock hack is that those three resistors ussually are not the same value, so if you end up driving all three colours at once the white balance comes out odd and the blue looks weak. I measured mine with the meter in ohms mode and just bumped the blue resistor up a size to even it out. Also, if that clicking pin turns out to run a piezo buzzer, keep it in mind, because wiring a chime into your own code is a fun bonus once the clock basics are done.