The "distortion" the LM386 is producing is "clipped" distortion, and probably looks something like this if the output were viewed:
That is a "hard" clipping at the tops and bottoms, and we perceive that as a "metal" type of tone.
When you overdrive a vacuum-tube amplifier, it also distorts the waveform, but the edges are much more rounded, and thus, pleasant to listen to.
Here is some more information on musical distortion, including a blurb about the legendary Ibanez TS9 "tube screamer."
The bottom line is, hard clipping will always sound "metal", and soft clipping will sound more pleasant. It is possible to achieve soft-clipping with solid-state electronics, but it's a little more involved than simply overdriving an input. DIYstomboxes have many schematics available for various types of distortions and other effects.
In general when building stomp-boxes you want high input impedance and low output impedance.
A passive guitar pickup usually has 6 to 15 kOhm impedance.
Typical values for inputs are 1 MOhm for the input. That's what has been traditionally used in the input stage of tube amplifiers. The value is so much higher than the output of the pickup that you will see almost no change in tone by plugging your guitar into this input.
You can go lower, but the lower you go, the more the input affects the tone. I wouldn't go below 10 times the pickup impedance (so 150kOhm would be my limit).
The LM368 has a input impedance of 50kOhm. If you directly connect your guitar to it, you'll load your pickups quite a bit. This results in a shift of the resonant frequencies of the pickup and some loss in treble. If you on the other hand connect this to the output of another stomp-box you may just lose some gain.
For comparison the good old Tubescreamer effect has a input impedance of 510kOhm.
Regarding output impedance of a stomp-box: There is no real standard here. Some effects are driving the output with a impedance of 10kOhm. That's fine and in the ballpark of what the pickup itself has. On the other hand some effects have the output as high as 100kOhm.
The schematic you've posted has a output of roughly 1Meg. That is a lot. The reason for this is, that there is a passive tone control in front of it. If the volume pot would be of a smaller resistance the pot would not only control the voltage but also influence the tone stack.
Overall I don't like the design. I would expect a simple buffer amplifier between the tone-stack and the volume pot. That would allow you to drop the resistance of the volume pot to something reasonable.
On the other hand I have never heard the effect, and the impedances might be just right for the tone. A good distorted guitar tone is the sum of all imperfections down the signal path after all :-)
Best Answer
The answer lies in supplying the plates of the 12AX7 and the 4558 correctly (adding connections to 12v in my case). The latter omitted from the original design. Rookie error, but it is now working well.