In the kerrywong drawing the L and C settings of the switch are mixed up.
You should have a parallel LC circuit there to the ground for the oscillator to run. When measuring C, Cx is parallel with the 1nF cap, and when measuring L. Lx is in series with the the 221uH coil. But you still need that modified (with Lx or Cx) parallel LC circuit to the ground.
As a side note, I have a similar circuit in a "secohmmeter", a flea bay purchase of a cheap LC meter kit, except it uses a 100uH coil and a 2nF cap for the reference oscillator.
With the pull-up resistor for the first comparator (the oscillator) being 1KOhm, I only got about 100mVpp signal at the output of that comparator. With 3.3K I got the full 0 - 5V signal.
Interestingly, a simulation of the circuit with 1K pull-up resistor would not start, but would run fine with 3.3K.
This probably has to do with the loop gain being higher with the 3.3k resistor.
According to the spec sheet of that comparator it could sink 5mA current, so the 1K pull-up resistor should be fine with 5V supply voltage, but in all their sample application circuits they used 3.3K!
Hope this helps,
Peter