So I am wondering if an LED in an amp gain stage would give a really smooth distortion?
Your examples show LEDs in the cathode circuit of a preamp tube. Ordinarily, you have a cathode resistor and bypass cap there. Each is doing the exact same thing, which is to provide a d.c. bias for the tube.
What is different about these, and why the mention of using an LED for a trem oscillator? You already know you can adjust the bass response of the gain stage by changing the value of the bypass cap. In other words, depending on the cap's value (and that of the resistor) it can approach a short-circuit at some high frequency while being a large impedance at some low frequency. At frequencies where the bypass cap is not a good-quality short-circuit, there is local negative feedback across the cathode resistor and reduced stage gain.
To get that good-quality short-circuit at very low frequencies, you need R*C to be large. The value of R is fixed by the needed d.c. bias of the tube, and that means you need a very large C. But even a 330uF cap across a 1.5kΩ resistor is -3dB at ~2Hz, and trem oscillator frequencies are around 3-7Hz. So we still might have insufficient gain from our oscillator with a 330uF bypass cap, and gain is critical to getting the oscillator started and running.
The solution is to remove the dependence on cap value by changing the bias method. You
could ground the cathode and apply fixed-bias to the grid, and get full gain down to 0Hz (because there's no frequency-dependent feedback). You might even supply that bias with a hearing aid battery or a AAA battery, if your desired bias voltage is ~1.5vdc.
Or you could use an LED. LEDs have a specific On-voltage, which is the voltage across the LED while it is conducting and lit up. The LED will draw as much current as needed to reach that on-voltage, and once lit will maintain that voltage even down to a fairly low current (a wide enough range for preamp tube use). You can take advantage of this characteristic, and use it as a free d.c. voltage pegged to the on-voltage of the LED you choose (different LEDs and different colors can have different on-voltages).
Like a battery, this bias method will maintain a solid bias voltage down to 0Hz, and give full gain from a trem oscillator.