Sure, we did similiar when the babies were young.
Do you really think miss stinky fingers is doing all that?
To me it sounds like they have a box of cloth wipes like I used with the babies, only mine were soaking in Dr’ Bonner’s liquid Castile, and it seems like they keep them dry before use. I would assume they have a covered pail to toss them in for washing.
If they live in an abundant water area, there wouldn’t really be a problem, as long as they were cleaning and disinfecting them.
My mil years ago owned a camp and they babied the old septic system. There was a pail by the toilet that caught ALL the toilet paper. NONE was to be flushed. That stinker was emptied and washed out a couple of times a week..