Vectored Approaches in a Non-Terminal Area
The more air traffic controllers there are available to work in a given area, the more focussed a controller can be in a given area. That's the whole background to ATC and "sectorization" of airspace. There is an infamous, if not hated quote from a controller in Moncton Center from years ago: "Give me two good men and an ashtray and I'll work the whole center." Tongue in cheek, of course, but his point of view is that some people were complaining about putting too much airspace together, not allowing them to watch the finer details and help pilots in their operation.
One of the requirements that ATC has is to watch all of his airspace, plus a margin for taking hand-offs and so forth. If staffing is low, requiring the "consolidation" of airspace into a smaller number of radar screens, each controller working must watch more airspace. This means that he has more airports to watch over, more enroute traffic to concern himself with, and must watch a larger range. Vectoring to final may become difficult, or at the very least inefficient, since it is harder to vector accurately when you watch a large range. Terminal control units, who vector regularly and use as little as 3 NM between aircraft, are very focussed and are always one a much smaller range than their enroute counterparts since their repsonsibilities often include a very small area and only a small number of busier airports. It's easier to determine, with accuracy, where a vector will take an aircraft if you're watching a 60 mile range than if you're watching 500 miles from side to side.
Also, even if the airspace a controller is watching is comparatively small, there are always a number of other duties that must be tended to. For example, in a busier enroute sector, there may be handoffs to neighbouring ATC units, incoming phone calls from other facilities, and coordination of flight data between controllers. None of the above mentioned activities can be heard on the radios, so it may sound like the controller isn't busy but he may very well be tied up. These activities all require the attention of the controller, as well as his voice. If he's talking to an adjacent ATC unit, he can't issue you a vector to final, and may end up "zinging" you through a localizer, which we all know wouldn't be a very efficient operation, and may be less than helpful at best.
So if a controller says he can't vector you to final because of workload or because, "another controller isn't around to help," there may not be much you can do about it. And he's not necessarily being lazy or uninterested in helping out, either.