This is one of those questions where the answer is "it depends." But fortunately, it usually depends on the hunter!
Outfitters can generally give you average distances for shots on properties they hunt. If these distances seem long to you, then just tell your PH you need/want to get closer. It will often be more work, but if that's what you're comfortable with, by all means insist on it. Most outfitters would tell you, for example, that shots on Vaal Rhebok tend to be a bit longer . . . but
@BRICKBURN took one with a bow (obviously at close range). Yes, it takes more work, and yes, it may reduce your odds of connecting, but hunting involves trade-offs. Any PH who won't accommodate you isn't really a PH you need to be hunting with.
Of course, there are some shots which will almost certainly be long - mountain nyala comes to mind - and here, insisting on close shots may mean very low odds of success.
Your maximum distance of 250 yards for a .416 Rigby is longer than I'm comfortable taking with mine, but if you're good with it, I think you should be able to get closer than that to just about every animal I can think of in South Africa. Even in relatively open country.
Trying for closer shots can - and for me usually does - result in a more exciting hunt. More busted stalks for sure, but also a more challenging and fascinating hunt.