It doesn't take two hands to hold and handle the detector during the search, so as a right-hander it stays in that hand.
Most recovery tool, except the Pro-Pointer, are smaller and lightweight, and certainly most recovered targets are also smaller and lighter than the detector so they also don't require two hands to retrieve (usually) and stick in the pouch.
In the early years of my detecting life I did use two hands and did have to clean the detector drip sometimes, They were hard rubber bicycle handle types back then. But by about '72 I decided I didn't like everything getting dirty so I only use my detector in my always clean right hand. The same with my pinpointer which in on the right rear of my gear belt. My Lesche Digger and pouch are on the left side and I use only my left hand with recover tools and target retrieval. The result is an always pampered detector and pinpointer (only the tip gets dirty), and only my left hand gets a little dirty. No mess to clean (other than my left hand.
So, if I were you, I'd use a little warm and lightly soapy water and clean the hand-grip, make sure it gets dry, then consider the 'tidy' approach like I use. I don't have to buy tape or other wrapping material or get sticky substance on the grip or rod, and when I decide to sell of trade a detector, even after a year or two, they look sort of 'dealer demo' condition. Oh, maybe once a year I might have to do a little light touch-up grip cleaning , but that's it.I'm just not abusive.
Just my thoughts.
Monte