You don't know how far away from the dip in the roof the leak is, only how big the indentation is. You could do a repair and still miss the leak farther out.
Without the piece of wood the tarp would go down into the indentation and you would still get a puddle. You want the wood or whatever cover to be higher so the tarp will be a bulge over the indentation and also reach out far enough all around to cover the actual leak(s) too. So it is all about the tarp unless you do the whole roof like they want.
So why do anything if it is working now? Does the campground have some sort of rules for unsightliness they will kick you out?
You might need a bigger wood covering and a bigger tarp, that's all. Or else do the whole roof properly and $pend, $pend, $pend.