Actually I would have to disagree. I believe that in a way, the treachery of Saruman helped Rohan to come to the aid of Gondor. This is an example of Success through Failure, or as Edward de Bono puts it, through the use of an intermediate impossible (
more examples here). Saruman provided a nearby threat and a focal point that Rohan could unite and rally against. True, Théoden was kept out of the action for a long time by Wormtongue's "leechcraft", but once he came back to himself he was, I believe, more powerful than ever.
His fury at Saruman, and especially the deaths of Théodred and Háma, made him a truly unstoppable force. This was the kind of leadership the Rohirrim needed to play their part in the Battle of Pelennor Fields. Similarly, I think that Théoden's death after he had played his part in the battle was important for Éomer. Not only did he emerge as the new king, but he and the other knights were spurred on to great deeds in the battle.
I don't believe that the Muster of Rohan would have been as quick or complete without the former threat of Saruman. Most of the army had already been put together for the defence of Helm's Deep, making it much easier to assemble the host to ride to Gondor. Much easier to motivate people to defend themselves against Saruman than to simply ride to the aid of another country. Gandalf and Co wisely used the momentum of the previous battles and just kept on rolling to Pelennor.
Sauron and his Captain were not expecting to have to deal with the Rohirrim in the siege of Minas Tirith. Once the cavalry of Rohan got through the wall the siege was broken almost immediately. It was believed that Saruman had engaged Rohan. If Saruman had been on the side of Gandalf, and Rohan's help had been a certainty, then better plans would have been made to deal with them. Consequently, I think that Sauron would have been better off in the Battle of Pelennor Fields without any help from Saruman. The Wild Men deserve a lot of the credit, as well.