One interpretation involves an analysis of new gear that was introduced around this era and in the prior few decades. I'm relying here mostly on my own understanding of the technology and the dates of introduction, as sourcing I'm mostly using audio/video documentation that demonstrates various capabilities.

Organ technology in this period, where a fully-electric approach become not just popular but affordable to the masses(roughly between the 1930's and the late 1960's), and there were frequent and exciting improvements and innovations, acted as a direct predecessor to the market for synthesizer instruments. The Hammond Novachord(introduced 1939) is one such example of how far back this technology goes - some might call it a synthesizer, since it uses electronics and a subtractive model, while others might call it a form of organ because of the kinds of timbres it has. Names that we still recognize today like Hammond, Wurlitzer, and Farfisa all saw plenty of competition - the best I can offer to convey a sense of the changes sweeping through the marketplace is the wiki summary page. In this interpretation, the 1960's saw a "perfect storm" where there was a generation of musicians who were more familiar with organs, plus a peak in their status as cutting-edge technology.

As more capable and affordable synthesizers appeared in the 1970's, they quickly overtook most of the functionality of the organ, as well as adding many more timbral possibilities which, naturally, were employed to create novel aesthetics. Notice that in the same period, electric pianos of various forms had also become commonplace and indeed starting driving jazz and funk sounds. The electric piano era lasted somewhat later, a little bit into the 1980's, up until digital synthesis techniques could adequately and affordably produce piano timbres.

Edit: Some might also say that the organ never really went away - we just renamed it!