I have never heard of someone doing this, does not sound like a good idea.
Probably the problem with the washing machine is if the water is too hot.
I rinse my tack off with a water hose and soap to clean it thoroughly and then let it dry well and oil it. Never had any trouble.
I've heard of people putting blankets that had leather reinforcements on the straps in the wash but never something like a leather girth
I don't think that's a good idea, water is the leather's enemy... You should use a glycerine soap after after nourishing with an adapted oil :)
I've never tried it but that definitely sounds like a bad idea