In TNG, only a fraction of the 1,000 personnel of the USS Enterprise is shown on screen. And I am quite sure that we do not see everybode of the Voyager crew (haven’t counted), even despite the crew was reduced to minimum capabilities early-on. Thus, I would reason that we see Tom Paris flying shuttles because “racing pilot” is one of his character traits that shall be shown in the series rather than because there is not enough personnel aboard the ship.
The more I think about the rules on Saucer Separation set up in the Core Rulebook, the more I like them: Apart from the fact that the rules would technically seem to allow to duplicate a Captain’s Yacht out of mid-air (I think everybody here agreed that this is to be ruled out by common sense) there are two legit interpretations on the rules, depending on your approach to “realism”:
First, you could reason that every section of the ship is a full functioning entity, operating entirely independent, but each on a smaller scale than the whole ship, unseparated. This is the approach I would favor, since there are enough NPC (in case of a Galaxy Class: about 1,0000; in case of a Nebula Class: about 750; in case of a Prometheus Class: 170 [Memory Beta]) to man the stations needed.
Second, you could also validly reason that, since capabilities are divided, they should, on each section, not exceed the maximum of the whole ship. You would need some sort of house-rule (again, argument would be common sense) to come to this conclusion, but I do not see any interferences with other rules etc.
So, basically, the rules are open for both point of views on “reality” or “plausibility”. It is up to the group, what aspect of ‘realism’ they want to emphasise.