> Inselbildung
That, and Temperature, explain why thermionic transconductance for current never reaches transistor levels.
Work your way to pg 166 (pg 18 in the PDF).
The main problem is that you can't get the grid close enough to get good leverage on the electrons, not with (1965-)producable stuctures. So the Gm varies roughly as square-root of current, and is low.
If you get the grid to the potential minimum, and avoid Inselbildung, eq 5.2-17 says 11,600/T Mhos/Amp. Taking T as 1,160 degrees (a bit hot), we get 10 Mho per Amp or 10 mMho at 1mA or 100 ohms or 100 milliVolts. This is a bit more than three times higher than a BJT working at 1/3rd the temperature.
So yes, a tube could be as good Gm/I as a BJT, if we could put the grid very darn close, without Inselbildung or stray grid emission, and find a good room-temp emitter.
It is interesting to me that it seems to have taken over 30 years to derive this result for the thermionic device, and Shockley got it for the BJT in less than 10 years.