The new 12AT7 tubes arrived (I ordered two just to have a backup) and with great anticipation I plugged one in… and found that the output waveform hadn’t really changed that much. Hmmm….
That had me staring at the circuit diagram and doing some more careful analysis, which basically showed that the device is working as designed. Then I entered the circuit into LTspice and ran a simulation – and sure enough the output waveform looked very close to what I’ve been seeing. Turns out that folks on the internet also mention the non-sinusoidal waveform that this thing puts out. So it has actually been working as designed all along. The design is basically a compromise as to what can be achieved with just two common tubes.
Once again here is the output waveform with nothing connected to the output, and the level turned all the way up. The frequency is set to approximately 600KHz in the AM band.

So what happens to this output if you actually connect it to a radio? I connected the output to the antenna input of my old Hallicrafters S-40 with about a 3-ft length of 300 ohm twin lead. Here’s what the waveform looks like at the output of the KG-650.

Sorry for the blurry image, but you can get the idea – there is definitely some 600KHz in there but an awful lot of harmonics as well. At the radio antenna inputs it looks like this.

Once again apologies for the poor photo. This is basically a slightly attenuated signal from the one at the output of the KG-650, which is to be expected.
The radio is actually able to tune this signal at 600KHz on the dial, and if I turn on the modulation in the KG-650 you do get a tone in the radio. Not a very pure tone, but a tone nonetheless. So it could be used as a rough alignment tool, which is why I bought it in the first place.
Now I need to decide if I’m going to leave it the way it is and just use it that way, or if I’m going to try to improve it. Decisions, decisions.