Funny little coincidence…
I hadn’t bothered removing this LED module from the side of my PicoCalc and ran your ZX-Picocalc emulator. The LEDs lit up (red, orange, yellow, green, blue) and stayed on while I was using the emulator. My guess is your Atari joystick support (a cool idea, BTW!) from those pins activated them, and that’s also why the 6th LED (which is white) never came on because the joystick pins didn’t use it!
Anyhow, I thought it was appropriate since this is the Spectrum, after all. 