I realize your version of the story doesn't include this, but I've often heard it as the restaurant giving the guy a permanent dishwashing job, and sometimes even having the guy rise up in rank and eventually come to own the restaurant. This version is even unlikelier. (For instance, if the restaurant thinks a guy hired this way is better than a guy hired through the normal application process, why do they even have an application process? And if you're in a situation where jobs are scarce, then jobs are valuable things and the restaurant should be able to be very selective in who it hires.)

I can't speak to whether there are any real instances of such a thing happening (the story is only vaguely familiar, I may not even have heard it before,) and I suspect it's more likely than not apocryphal. But the answer to "if the restaurant thinks a guy hired this way is better than a guy hired through the normal application process, why do they even have an application process?" would be "because a single specimen does not invalidate a selection process that deals in generalities." The application process is an attempt at sorting prospective workers to select those who are most likely to be valuable to your business, but that's not to say either that some duds won't get through (people routinely make it through application processes only to be fired for poor performance, after all,) or that a less stringent selection process cannot induct good workers.