I also use paper towels for handles.
As a teenager, I was the custodian of my church (age 13-18) so I cleaned mens and ladies bathrooms daily. [it was an active church]. I learned women and men are equally messy and dirty. Women’s messes were a little messier in the shredded toilet paper everywhere and the men’s were urine around the floor.
In everything else, equal. Oh, wait: women’s room mirrors often had lip marks on them. That was annoying to clean.
Ladies room *did* have the extra feature’ of the tampons and maxipads… and sometimes they didn’t use those, which left a *different* kind of mess to clean up…
…but I’ve always had a strong stomach so I just did it. Never complained or said anything. You can’t help what you have to do in a bathroom.