Separating floor bath mats and tea towels

My housemate and I just got in a bit of a disagreement because I said I’d prefer the bath mats (we have ones that get stepped on all the time not just after showers, ones that sit in the bathroom and get walked on and also one that goes around the toilet and gets stepped on when people use the toilet) not be washed with the tea towels and cloths that touch our dishes. I prefer to separate them in to bath mats and dirtier towels then do a separate load for tea towels, dish cloths and hand towels. She thought I was ridiculous but my logic is you wouldn’t was dirty floor mats with your sheets or other clothes so why would you with stuff that’s going to be touching our dishes and hands? I know it all gets washed in the end but I still feel iffy about mixing that stuff. What’s your take?