Fasting seems insane to those who don't know anything about it and haven't experienced it for themselves. They always see it as starving. Starving and fasting are completely different. Plus, you aren't starving your body, you're telling your body to use it's own fat stores for energy. That's a good thing. And as long as you are feeling good and feeling healthy while doing it, that is what matters and tells you what you are doing is working.
I understand your wife having concerns. I'd have her do some reading on the subject, to set her mind at ease. I highly recommend "The Obesity Code" by Dr. Jason Fung. It's a great book, and covers everything about what makes us fat, and what does and doesn't work to lose fat. It covers why certain diets work at the beginning, but then ultimately fail and what will work as a sustainable diet. It goes into great detail, but is easily readable. It's what really convinced me to do fasting in the first place. Anyone who wishes to lose weight, should really read this book.
Once I did my first 3 day fast, and saw how I felt, it was a game changer. Over the summer I was at the 24hr mark, maybe longer, and did a 33 mile bike ride, and never ran out of energy. I just finished a 6 day fast (first time going that long). It was amazing. I still have energy and I feel great. I had a desire to eat, only because I wanted to eat something, not because I was hungry for food. I can totally see myself going on longer fasts than that.