Could a resistor work as well as a signal diode in this level-shifting circuit?
I don't think so. For one thing, the WS2812 chip receiving 3.3V data may actually work. It may not be reliable, but you can't count on the LED's to stay off nor on or in any consistent state.

So then if the first chip changes output state of its LED's, then the supply current will also change. If the supply current changes, then the current through the diode will change. If you substitute a resistor instead of a diode, with the potentially non-constant current, you won't get the voltage drop you are trying to get. You will get a variable voltage drop that depends on the state of the LED's in the first chip. Also, these chips have internal switching. This means that even if the LED's are in a consistent state, the current will not be constant. So, all around, it is not a good idea to try to use a resistor.

In my opinion this circuit is really crappy. I would suggest you use a level shifting buffer if you want reliable operation. The sacrificial LED makes no sense to me.