Although not frequent, and usually not for very long, databases will experience problems and just not be able to be connected to.
An error message of just "Unable to complete this request due to an error. Error has been logged." just doesn't seem too friendly to visitors.
Perhaps I just haven't seen this (or discovered how to do it), but if it isn't currently an option (or a hook) for this, it'd be a welcomed thing. (If there is a way to do this, I apologize...it's sometimes hard to come up with the proper search terms to find results.)