On the subject of hygiene, horses like a good roll in the dust, as do dogs. The Navajo, as an example, who live in an area where water is almost non-existent without digging and waiting for a small seep through the sand, didn't make daily treks to the bottom of the Grand Canyon to wash in the river. They did just like horses and dogs do naturally, they took dust baths. Sometimes these would follow a bit of time in a sweat lodge, but other times it was just how they bathed.
I have washed my hands in dirt, when I didn't have enough water to waste. It will get rid of the smell, and take away any particles that may be clinging to you. If I find water, I will use it, but dust or sand is better than nothing. This is not trying to be PC, just part of growing up a child of the desert. If you find yourself in the desert unprepared, it's what you do.
Being prepared is a good thing, though, with TP and handi-wipes; but I would imagine the PC thing would be more like I described above.
As far as MM bathing? I remember an old joke about a cowboy getting ready to take a bath in the fall and remarking: "No wonder it was such a hot summer, I forgot to take my long handles off last spring."
Bathing was probably something you did to relieve yourself of who lice, more than having a "fresh as an Irish Spring" feeling and smell. For re-enactors, however, I think cleanliness and hygiene are important, as you are indeed among the public and they wouldn't really appreciate the PC smells.