The problem with If it's not broken, then don't mess with it!
is that it can become a management mantra that leads to premature code ossification. ("Ossify" means "to change into bone." In the context of code, it means hardening against change.) You want
to be able to "mess" with stuff in a controlled fashion. That's what "Refactoring" is all about. If you don't refactor, you end up with Big Balls of Mud
The last thing you want is some pointer-haired manager wandering about doing the "It's done! Ship it! And for diety's sake Don't Touch It!" That road leads to a place you don't want to go.
I'd rather see this written as
Don't mess with working code unless you have unit tests!
Otherwise, how can you be sure?