My personal recent example is a wizard in a web app built as a single major component, with some subcomponents, that goes forward and backwards through 3 different major flows depending on use case. It's been somewhat tamed over time, but we still find cases where an error on step X gets us into an unrecoverable state, sometimes with actual longstanding data impact or where closing & reopening the modal doesnt get the user back on course. With a state machine, I feel, things would be far more predictable.
Having some actual Computer Science in your coding can be a huge boon.