Thank you.

I do agree that Eowyn possessed the qualities of a good leader. She was self-sacrificing, and loyal above all else. Had she stayed, though, I believe she would have always regretted it.
Besides that, LotR would have had a slightly different ending. The Eowyn/Faramir relationship would not have blossomed, and neither of them would have been able to grow and heal as they did, emotionally. Alright, I'll go put my E&F shipping fanaticism away for now.
I like what you said about Eowyn growing with the task. I suppose it would have been more of a "train-as-you-go" type of experience for her, through the ins and outs of governing and warfare.
Theoden had so much to think about at the time. First and foremost, there was his country. Then, there were his loved ones. Leaving Eowyn as stewardess seemed to answer two dilemmas at once. It would be easier to leave Eowyn as stewardess than it would be to have her in the very heart of danger, and the people loved her and were loyal to her. So, she would be safer and more protected, and the homeland would be strong under her. Obviously, these were the things most important to him, and must have most influenced his decision. But it was Eowyn's spirit that he seems to have underestimated, or perhaps mostly left out of the equation. She wanted to be the most help she could, despite the cost to herself. It was her love and loyalty that drove her out to war, while it was Theoden's love and loyalty that would have kept her at home. That spirit in Eowyn was not one that would easily wither and die, especially not when she had already made up her mind and steeled her will.