Chestertons fence is basically a prior: Things are the way they are for a good reason. It can be more or less applicable in different contexts, and I'm not sure I would give give bureaucratic procedures such a benefit of the doubt. On the contrary, I think that any departure from doing something in the simplest possible way has to be accompanied by strong evidence.
Here's my take. Any requirement or bureaucratic procedure is meant to mitigate some risk. You must understand what that risk is before removing that process step.
Many times that process step is just arbitrarily layered onto an existing process. If you do that enough times we get the inefficiencies that we tend to think of when we say "bureaucracy." That creates a risk in itself, namely a productivity risk. But that stems from my earlier point: when process steps are just layered on without thinking about the second-order effects, it creates a bad process by incurring outsized risk. But the converse is also true: removing process steps without understanding the rationale behind them can also increase risks.
You seem to equate bureaucratic steps with needlessness. I'm not sure I agree. I tend to think they are there for a reason, but possibly just implemented poorly. Same goes for decisions to remove steps from a process.
