With your response on how Edmure should handle the defense of the Riverlands, it assumes that both Tywin and Jaime have to cross the Red Fork, in a reverse battle of the Fords. However your mountain pass answer would have Tywin come from Deep Den, loop around and hit the defenders along the Red Fork from behind at Pinkmaiden. Which would rupture the entire defense and leave the Riverlands troops strung out in a line from Riverrun to Pinkmaiden, with Tywin and Jaime hot on their heels.
That is a trenchant critique, and I’m glad you asked it!
Even in that scenario, I would still maintain that it’s a better strategy than sending 4,000 men to get butchered uselessly at the Golden Tooth and theng, waiting at Riverrun until you get overrun.
For one thing, it gives you 4,000 more men to hold the Red Fork, which means extra men available to go out scouting for Tywin’s army so as not to be taken by surprise, extra men available to defend Pinkmaiden aaginst Tywin’s army (preventing him from rolling up the line), you still have the option to throwing in the reserves to push Tywin’s second army back, and you have at least a decent chance of getting the army back to Riverrun in good order. Certainly much better chance than in OTL.
For another, it’s still going to inflict more casualties on the enemy than OTL: Jaime’s army still needs to cross the Red Fork and Tywin’s army is stil going to have to assault a well-defended castle. And wearing down the Lannister forces also means slowing them down, preventing that lightning march across the south that knocks out so many castles (including Riverrun).