Okay, I shorted pins 1 and 3 on Q3. Still nothing. It beeped, but no display.
I tried it 3 times, but now it's not beeping and there's two dim orange lines running horizontally across the display on the inside. It does it even if I'm not shorting Q3 :/
Q3 controls the filament voltage in the tube as well as power to the VFD chip. The missing 22 ohm resistor was preventing the filament voltage from getting to the tube, but it seems we have other problems as well.