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[LOTR: The Two Towers] Théoden listens to Aragorn and Gandalf and rides out to meet Saruman's army instead of evacuating to Helm's Deep. What changes?

In the movie, Aragorn and, more importantly Gandalf, urge Theoden to take the fight to Saruman head on. Instead, Theoden chooses to evacuate his people to the safety of Helm’s Deep and weather a siege instead of risking open battle. Gandalf especially seems to believe this is a foolish choice and rides out in a huff to go find Eomer and bring him and the Rohirim back to fight. \*But…\*didn’t Theoden make the right call? Had the remaining men ridden out to fight Saruman’s orcs and Urukai in battle, while they would have maybe had an advantage in mobility (depending on how many warg riders there were), they would have been vastly outnumbered and Edoras would be completely unprotected. They would have been even more outnumbered because the old men and young boys who fought at the siege would have been next to worthless in mounted combat. Edoras is a hillfort with very few defenses for the main hall, let alone the farms and villages around it. So even if the main part of Saruman’s forces engaged in the main battle, a small contingent could easily destroy the city and kill all the civilians. And had Theoden sent the women and children out to Helm’s Deep while taking his army to meet the orcs head on, then the warg riders would have slaughtered the civilians on route. Gandalf still would have taken the same amount of time to reach Eomer and bring him back, so wouldn’t have tipped the balance any more than it did in the actual battle. The men still might have won the battle, but it would have been at the cost the lives of most or all of the women and children. Gandalf acts so self-righteous and pissed off when Theoden decides to evacuate to Helm’s Deep, but it seems to me that Helm’s Deep was the only option Theoden had where he had both a chance of winning the battle and saving most of his people.

Edit: I'm talking specifically about the movie. I don't remember if it's different in the book. Also, I forgot about the elves. But I don't see how their presence on an open battle field would change the outcome anymore than they did at the siege.
gpt
Evacuating to Helms Deep provided one single concentrated point where all of the military strength Theoden could muster was located and it made Saruman's best course of action a simple and direct and overwhelming strike at the point to eliminate Rohan's leadership and available military assets in the region making an organised defence next to impossible. But if Theoden had gathered an army at Edoras he could have amassed a force similar in size to the army he took to the Battle of the Pelennor Fields and sending out patrols and detached companies (like the one Eomer lead earlier in the film) from Edoras to patrol and skirmish with uruk-hai raiders to protect the villages most of the population lived in, utilising the wide open spaces and the Rohirrim's greater mobility to generally win such battles. That generally would have made it an attritional war of movement that would have worn down Saruman's forces over time, but it would meant more raids and destroyed farms and villages, and more casualties among the civilian population and a stready stream of battle casualties of the length of the war. Saruman might have tried to directly attack Edoras (or another important castle or town) with a large army but that army would have been intercepted en route and either harassed or engaged (with similar or greater numbers) in terrain the Rohirrim are superbly well suited to fight on.

Generally speaking fighting out in the open more aggressively would have been a more certain outcome for the war but would have taken a greater toll on Rohan while defending Helms Deep invited a risky battle that would have decided the war quickly while surrendering the initiative to Saruman entirely, if he wanted to Saruman could have set up fortifications on the roads to Helms Deep and starved out the fortress while the bulk of his army ravaged the rest of the country.