So I did an experiment for you. For science.
This guy posted on Reddit about how he hasn't used soap and shampoo for a year. I've heard similar ideas before, so I decided to give it a try.
For over a month, I showered only with water. I never stopped feeling unclean. My hair had lots of dandruff. My scalp itched. Scratching it felt tacky due to the oil on my hair. I cut my hair to 1/3 inch, and it wasn't much better. A month went by. No improvement.
I used soap & shampoo again today. I finally feel clean. My hair smells good. My scalp isn't itchy.
Maybe not using soap and shampoo works for people who shower 3 times a day and swim in chlorine regularly, but it didn't work for me.
So scratch that theory.