Yeah, you could do that. I suspect the only reason most people don't is that it requires more planning; you need to go through two starters before you're ready to pitch, and you need to be thinking ahead to future batches - i.e, you need to know you're going to use the yeast more than once.
And washing isn't that difficult. Just harvesting the yeast after it finishes is fairly simple; washing it just takes a few transfers between sanitized vessels.
One possible advantage to washing yeast, I suppose, would be that you are getting yeast that's been active for longer. If you're planning on using it again soon, that may mean the difference between 1-week old and 3-week old yeast. If future batches are farther out, that may make little difference, though.