I tried exactly this when I was working from home for a while. The hair worked out wonderfully - better than any product I’ve ever used.
But the BO? It... well, my wife called it a manly musk. She gave it a doubleplus good at home, but i was not allowed to leave the house like that. Even I could smell myself.
I tried adding some probiotic body lotion into the mix, but it didn’t help at all.
And what happens if you accidentally get soap on part of your arm or leg?
Nothing happens if I get soap elsewhere, I just don't intentionally soap outside those regions daily. In my experience arms, legs torso etc aren't particularly dirty on a daily basis and the skins seems more mostrurizied and healthy if I just use water on them. YMMV depending on skin type.
This isn't a case of me not smelling it. I just have another cold shower whenever I need one. But generally speaking most exercise is fine.
Not to mention those of us in customer-facing roles, who can’t really afford to hope we’re not offensively ripe that day.