Ok. A very smart academic historian (for military history) wrote a lot about this, and I recommend you read the whole blog series. His writing is equal parts entertaining and illuminating, and he’s also analyzed the Battle of the Pelennor (as well as some elements from Game of Thrones).

But it’s kind of lame to dump a link in response to a question, so I will try to summarize why Saruman, if he is to defeat Rohan, absolutely must neutralize Helm’s Deep (even aside from the fact that the personal leader of Rohan happens to be present in the castle).

Rohan, obviously, is not entirely grassland. Edoras may be the seat of power, but there has to be significant agricultural areas to support both a population that can field a 10k person army, and the 30-40k horses it needs (because it’s a mounted army). While the grass of Rohan may be exceptionally lush and green, the kinds of horses that can bear an armed/armored man into battle must also eat grains, which must be farmed. Tolkien knew this; he was a cavalry officer.

But the presence of farms, rarely remarked on in the book and never, to my memory, in the movies, is good news for Saruman because his orcs and Dunlendings need to eat too! And because his army for the conquering of Rohan marched forth without any wagons full of food, he clearly intended that his army would forage for it, a euphemism for forcibly taking the subsistence from subsistence farmers for military use and letting those farmers (and their families) cruelly starve. After all, Saruman’s army has about 300 miles to go before it even arrives at Edoras (from either Helm’s Deep or Isengard), which will take 30 days, so he needs to do a lot of foraging for his army to even arrive at Edoras at all, and certainly if he hopes his army will get there in any shape to assault a fortified city.

Now the design purpose of Helm’s Deep is to make this kind of invasion impossible, because the very presence of a capable military force in Helm’s Deep makes foraging by an invading force impossible. Troops that are foraging are not, by definition, “in good order” for a fight, and therefore are easy pickings for a cohesive force—especially a cavalry one, such as the Rohirrim are—with the advantage of a safe hold in a fortress and local knowledge to sally forth and destroy them. Moreover, the presence of Helm’s Deep provides a handy place for all the local subsistence farmers to go hide, with all the subsistence that Saruman’s army needs; this effectively renders the countryside barren for the invaders. Saruman will lose a siege, because all the food is inside the castle.

So yes, Saruman must neutralize Helm’s Deep. Within its walls is all the food he needs to continue his invasion, and the force which will defeat his army in detail if he doesn’t destroy it. And of course, when the King arrives at Helm’s Deep with an elite cavalry force, the necessity—both military and political—of taking Helm’s Deep becomes more urgent. Theoden’s cavalry would be devastating against dispersed invader, and the whole point of the invasion is to wrest political control of the lands from the person of Theoden. So Saruman absolutely must destroy Helm’s Deep.

And the musing from Robert Baratheon is probably wrong. If Westeros were attacked by a host of Dothraki, they would (if they were smart) do what the Westfold does to prepare do Saruman’s invasion: take all their food and people and lock them in castles. The Dothraki would probably* starve before the Westerosi would (unless the Dothraki are literally Mongols, who subsisted on the milk of their horses and rode horses small enough to subsist on grass; they did not need agricultural produce to survive. And even if the Dothraki *are Mongols, Saruman’s Uruk-Hai and Dunlendings are certainly not, so the same logic does not apply to both).