There some text around the net explaining another designers issues showing the same problem. Using any Xtal controlled transmitter will show the same Fming issues if viewed on an SDR.
Quote from the text follows....
I tried various means of making an oscillator for 3.615MHz along with a VFO to cover that frequency but I couldn’t get one stable enough. It varied only a few Hertz but that was enough to cause asymmetric sidebands (one sideband output higher than the other). A crystal oscillator would be the ideal solution but as with most components these days, they are not readily available and more than one will be needed for other frequencies. There is a UK company that can make these but they will not be cheap because they are not mass produced. I tried ‘pulling’ a ceramic resonator to frequency and although I could cover the required frequency, it was varying by a few Hertz. This is not detected on air because it’s only a tiny fluctuation but on SDR it can be seen as asymmetric sidebands on the AM transmissions. This shows on average a 6dB difference in levels in the sidebands at low frequencies. It’s not quite so noticeable above 1kHz but it’s there and because it’s there it will be noticed and will commented upon. Fig. 2 shows perfectly symmetrical sidebands at 300Hz. Before SDR no one would know or even care but with many operators using SDR they look for any imperfections! So, a stable oscillator had to be found. I tried a GPS and etc etc...
I find that explanation a bit problematic:
1) The author of the text you lifted (without attribution or reference, BTW) is talking about a few Hertz of variation. Chris' image shows much more than that.
2) Back in the day, every ham used to use crystals exclusively. Many didn't have VFOs until the early 1960s. I can totally believe that crystal generated signals hop around a little bit. How much? Who knows and if there was only a few Hertz of variation on a CW signal generated from a crystal, it might not have been noticeable in 1950, especially with all the other key clicks and whatever going on. But more than ~20 Hertz variation would have been very obvious on a CW signal. So you are saying that dudes back in the day with their FT-243 WW2 military surplus crystals were FM'ing ~1 KHz all the time? Pffft.
No, I'm sorry, but no one would have been able to make a CW QSO if that were the case and RST 599 reports would have been unheard of.
To be clear - I'm not saying that crystals are perfectly stable; I'm sure that they are not. However, I don't think that the fact that the reference was a crystal in your old transmitters explains all the FMing that Chris shows (and others have reported). So I'm suggesting that there might be something else going on in addition.