Women nearly doubled the workforce within a couple of generations, but the sky never fell. And that shift was largely precipitated by technological advances, such as the washing machine, reduced infant mortality, and birth control.
In as much as doubling the workforce is similar to halving the number of jobs in pre-existing categories, I don't see the sky falling. Like with women joining the workforce, the shift to more automation will stimulate new kinds of labor demand. And it's similarly likely to make education even more important and relevant.
That said, women joining the workforce is perhaps the single largest contributor to long-term stagnant wage growth. Such shifts are unlikely to prove the end of labor, but they do create significant displacement that requires serious attention. Nothing about the "free market" guarantees that the economy and labor system will adjust quickly enough within timespans relevant to maintaining the well-being of the individuals displaced.
I don't really think there'll be anything particularly challenging about the next shift. In simple terms I don't see how it could be much more problematic than the addition of women to the domestic workforce immediately followed by hundreds of millions of Chinese laborers joining the global industrial economy. We just don't appreciate how disruptive those historical events were because, despite the weak U.S. response (as compared to Germany), even unassisted the labor economy did a decent job of treading water. It's quite amazing in retrospect.
Theoretically we could learn from our past mistakes and not only adjust but come out ahead in terms of income growth and equality. OTOH, we could not learn from our past mistakes, and our current issues would only be compounded. On this I'm much more pessimistic. When the electorate and political culture still pine for a labor economy that is 2 or 3 epochs in the past (i.e. ancient history!), that doesn't bode well for their ability to commit to the demands of the future.
Even worse is the fact that the intellectual class seems oblivious that history is merely repeating itself. The books about robots and labor literally have almost identical titles to books I read in the 1990s in college seminars discussing the implications of women, China, the information economy, and of course the rise of robots. Some of them were written in the 1980s. One famous book, "The End of Work" (Jermey Rifkin, 1994) could probably be re-published today with nary an edit. People would find it just as timely and insightful as they did then, and it'll prove just as irrelevant 25 years from now as it is today. We saw neither the dystopia nor the paradise he envisioned, but rather (as always) a slow, systematic, and largely hidden reorganization of the labor economy that became evident, if at all, only in hindsight.