Did you forget about the random part?
Even with a micro, the random part would be hard to mimic. Controllers running set routines come up with set answers when algorithms are run. The best way to get a random number would be running an asychronous clock and having the user push a button to read the value. Since the user can't push the button at set intervals with a fast clock, the numbers would truely be random.
Maybe a pair of asynchronous clocks feeding internal timers and an algorithm that mixes the two resulting numbers when the timer data is read. Having the user push button the start instead of under program control would ensure randomness. Then a modulo six routine to extract a 1 of 6 position from that.
We had a hooty hoot owl project in high school electronics that blinked a pair of eyes (neon lights). We scavanged three or four kits to string together 8 lights, but they fired in random sequence. Just a simple RC circuit. We worked two days to try to synchronize them, then I hit on the idea to feed the output of the last back into the first and viola! It had two 75K resistors feeding the two neon lights, and a capacitor between them after the reistors. I'm not sure if it would work with lamps instead of neons. Maybe I'll play around with it to see. I seem to remember it used 100VDC but I'm not sure (it was the 70's after all).