How does the spectrum compare?
Both Johnson and Schottky (shot) noise are white, yes.
The article linked earlier showed a measured noise voltage of 0.26uV for a red LED at 5mA.
Assuming that voltage is measured over a 20kHz bandwidth that would be equivalent to the Johnson noise from a 200 Ohm resistor at room temperature.
So it seems that if the cathode resistor you would use is higher value than 200 Ohm using the diode would be lower noise.
I haven't read the article but when it comes to noise measurement, most people get it wrong. Measuring the noise
voltage can get really confusing when you're trying to compare with something like a resistor, because you have to take into account the different effects that each would have in an AC circuit, to get a fair comparison. In a tube stage a bias resistor causes degeneration, an LED doesn't, for example. Voltages can trick you, but currents tell the truth.
An LED may well produce a lower noise
voltage at the cathode while still injecting a bigger noise
current into the tube (because the LED internal resistance is low). This will still usually give you a worse overall noise figure, because you're injecting something like six times more noise current (say), but you're probably not getting six times more gain compared with using a resistor.
The LED does give you the option of running at higher tube current without changing the bias voltage, which a resistor can't do, of course. But that's not comparing apples with apples. Even if you have a lucky situation where a bias LED nets you a better noise figure than a resistor somehow, adding a bypass capacitor to the LED will still likely buy you a few dB improvement. But you could have added the capacitor to the bias resistor too so...
The more I learn the more I have become convinced that it's hard to beat a simple, resistor loaded, RC-biased triode stage. Every time you think you've made an improvement somewhere, you find something different has just got worse, and your grandfather is winking at you. Anyway thanks for coming to my Ted talk.