Wondered about the oils bit. But been doing it this way for years and no problems. Always give the leather a soaking and wringing before use when it is dry.
we used to clean our chamois and flunky's- fake chamois by laying on a pre cleaned tile floor and jet wash them clean, used to play a game were you had to flip the chamois over after cleaning 1 side and lay it out flat but only by using the water jet