Butler was basically reiterating a cyclical view of history; this has been popular for some time, though it was really formalized in the 19th and 20th century through works like Oswald Spengler's The Decline of the West (1918), and with respect to the United States in particular through books like Strauss and Howe's Generations: The History of America's Future, 1584 to 2069 (1991). As the title implies, it is based heavily on the idea of generational turnover and change.

As a literary trope it has seen wide success in fiction and science fiction, for example in Isaac Asimov's Foundation series (where the practice of "psychohistory" is essentially based on being able to predict such things) and Frank Herbert's Dune series.

As far as historical weight goes...it depends on your opinion of historical cycles. It can encompass such things as business cycles, dynastic cycles, political-demographic cycles, etc. The main issue with calling them cycles is that it implies periodicity - as Butler here says "every three or four decades." However, human populations and their attendant activities and political structures tend to be a lot more messy and complicated than that, so the "cycles" usually aren't that neatly defined - really only being apparent after they've happened, and even then not always by a quantifiable standard.

For example, WW1 is normally said to have ended with the Treaty of Versailles in 1919, and WW2 to have started 1939 with the German invasion of Poland, a period of only ~20 years, versus the 30-40 specified by Butler - how would that work into the theory? Does it work into it at all? Once you have the idea of history as a cycle, you can start looking for dates to fit, and given how complicated the world is probably find some event that fits. At that point you're basically assuming the result.