This just hasn't been my experience but I suppose it's possible if your team is determined enough to write bad code. I'd still wager a bungled state machine is probably fairly easier to fix than a bungled mess of branches, but I've never seen such a thing.
I actually use passage of time as a factor in state machines all the time on game dev projects. It's pretty simple, just store a start time and check against it. I don't see how "ten seconds have passed since entering state A" is a more difficult condition than any other to draw or model.
In my experience Ye Olde Web App backend services tend to be particularly bad with time because so much is generally done in the request/response model.
For business-logic reasons, where I've generally seen it fall apart is when things go from fairly simple things like "after six months of inactivity this account is considered idle" to more complex interactions of timers and activity types. "Move fast and break things", "just get to MVP" attitudes rarely have the discipline to formally draw all the distinct states out as the number of potential states starts to exceed a couple dozen.