JCinPA
The Lather Maestro
The thing you are trying to get rid of is mainly mineral deposits from harder water, like soap scum. If you have soft water, you probably won't have a lot of mineral deposits.
People that wash their hair with plain old soap can have a similar problem. Their hair can become flat and dull. They are often advised to rinse their hair with a weak vinegar+water solution after washing it.
Interesting. I learned something. I guess our water is not too hard, never cleaned a brush, ever.