The only politician's writing that I've read is Ron Paul (Besides an occasional Op-Ed). I'm not talking about speeches, I mean enough writing to actually get a sense of what they really stand for. You can't expect big politicians to do this. They already have an audience and their own techniques for reaching their constituency.
But I think consistent writing is a way for smaller politicians to differentiate themselves. They can enunciate their position and platform. It's a more honest way to engage their audience. You can't write long-form without thinking.
David Perell says that your writing is your resume. Your writing is a portfolio of ideas. Can you identify the ideas and platform of any politician? If a politician wrote, you could exactly what they stand for. Without the physical proof of the written word, politicians cater to whoever is listening to them at any given moment.
When the tide goes out, you see who's swimming naked.