Why won't my JFET Hartley oscillate
It turns out that the original Popular Electronics schematic had an error where R3, the drain resistance, was 1000 times bigger than it should have been.  It was printed as 270k when it should have been 270.

However, even with this error corrected, I found that the circuit performed poorly in reality.  Biasing was difficult and the resulting sine wave was distorted.  A much better circuit includes a diode at the gate to shift the DC level and bias the FET into its linear region.  The details of this circuit (along with a description of the diode function) can be found in this stackexchange question.

The issues I had with fake MPF102s, whilst annoying, had no bearing on the issue or its solution.

Finally, 100% of the solution credit is owed to Andy Aka.