Its not disgusting, animals don't clean themselves now do they. My parents are Indian and I am British and I never shower during my period. I do however wash my face, legs and armpits. When I get my period I use a tampoon and pad at the same time and change the tampoon and pad once the pad gets quite full.
I was brought up like this and never found a need to change my habits and I do wash other parts.
Well even in India most people do shower during their periods but some chose to not shower. Even if you google it you will find that there are a good amount of people who don't shower. Most people don't recommend it but I have always done it.
You are right it does smell but only when you remove your clothes.