There is some evidence for debt cycles e.g. see works of Minsky (1987), Suter (1992) or Dafermos (2017). However, there isn't distinction being made between short and long-term debt cycles like in the video. Also economy does follow cycles around its long-term growth path but those are business cycles not debt cycles.
In economics, what the video author calls 'debt cycles' would be considered business cycle, since the author defines these debt cycles as fluctuations around steady growth path of an economy and in economics we call that business cycle (see Blanchard et al Macroeconomics an European Perspective pp 183).
There are some theories, like the theories of above mentioned Minsky, that business cycles are caused by instability created by overleveraging. There is support for such idea in some cases. For example, Great Recession of 2008 or Great Depression of 1929 are claimed to be an example of this as excessive leverage played role in both crises. However, I do not think there is any serious economist who would believe that all business cycles are driven by debt (e.g. oil shocks of 70s, Covid19 or Russo-Ukrainian energy shocks are clearly not driven by debts and would cause severe recessions even in counterfactual world where governments and households are not much leveraged). Furthermore, I do not think most economist would even say that leveraging causes most crises, rather often leverage is correlated with business cycles, but sometimes it can cause recessions.