So, after I started troubleshooting why it was behaving as if the Mode button was always pressed, I found out that that actually was the case; the button I was using was broken, and was always a closed circuit. Once I removed it, suddenly the Nano shut off, and since then I haven't been able to get any response out of it; not even the power light comes on, and it doesn't show up in dmesg.
Rest in peace, Nano. The last couple of days were brief, but enjoyable.
Since the only Nano I have has now passed on, I needed to find something else to drive this whole project. I could have pulled out another of the PICs I have in store, but honestly... they're more of a pain than I want to deal with right now. So instead, I dug up this tiny one:
It's apparently a Digispark clone, and it was given to me a couple of years ago by the same friend who donated the Nano. (thanks, Rusty! I couldn't have done this project without you :D)
Turns out, if I'm not tied to driving an LED display, I can do everything on 4 I/O pins, and still have a serial connection for doing stuff like display, if I get creative. (maybe I'll make a companion app for my Raspberry Pi Zero or something) Sadly, it can't do the serial monitor over USB like the Arduinos can, but there's other ways to get output.
If all else fails, I have an MSP430 (also given to me by the same friend), and I also still have a new "Pro Mini" Arduino clone coming... which is basically a Nano without USB.