Elimination diets work for some people. It’s estimated 2%-20% of people have some sort of food intolerances. There’s nothing quacky about it.
I’ve found it useful in my life as well although I don’t need to go to something as extreme. Generally you eliminate almost everything and slowly introduce foods to see what you can and cannot eat.
I had major issues eating wheat. A lot of my health issues began in Italy because the started a diet of food made from wheat. I had to spend 2 weeks in the hospital with kidney issues and actually had really bad dermatitis on my face.
As soon as I cut out the wheat, my kidney went back fully and my paralysis went away. I also no long have dermatitis flair ups all the time. The last time I had it was when I ate a whole pizza by myself.