There's no way the average person is washing their grocery bags in the clothes washer.
Sounds like that could coat your washer with microplastics that might end up in your clothes and against your skin all day. That may not be the best idea.
Pretty much any synthetic fabric, which includes most fitness wear, is going to fill your laundry and washer with microplastics. However the real problem occurs in the dryer, which heats the stuff and produces dust. Running some relatively solid plastic bags through a washer (only) is probably 999 on a list of 1000 things to worry about regarding microplastics in your home.
As far as what “the average person will do,” I’ve never personally had a hygiene problem with reusable plastic bags that couldn’t be solved with a sponge or a Lysol wipe in 30 seconds. But if the OP is really suffering with large numbers of dirty bags, a gentle wash with detergent is the simplest and most effective answer. At a certain point, it feels like this discussion is more about preferences re: reusable bags and less about trying to solve problems.
Sounds like that could coat your washer with microplastics that might end up in your clothes and against your skin all day. That may not be the best idea.