Yes, using the trailer outdoor shower hose is a good way to reduce the amount of waste water going into the trailer grey water tank. My wife and I prefer taking outdoor showers — including at our house where we had one added as part of a remodel — but in a campground there are of course privacy issues. We solved that with one of theseIn general, the limiting factor will be our 24 gallon grey water tank. Dumping grey water is illegal in national forests, and not advised in general. I can take outdoor showers but not everyone in my family is as, uh, amenable to that idea.
https://www.amazon.com/gp/product/B01AT3SZ36/ref=oh_aui_search_detailpage?ie=UTF8&psc=1
However, as @ohmman notes, there are other reasons not to take outdoor showers in campgrounds: scented bath products such as shampoo and soap may attract bears.
Thanks for the very informative trailer towing trip report!