> Can you use the cotton bag 173 times? Cotton wears out over time.

Yeah, comfortably. Shopping bags don't actually get much "wear" in each active use period. The main issue will be the handles but good stitching will solve that.

And it'll go in the washing machine with your clothes (though you might want to tie the handles together).

I have a Lidl bag that is already into the sixties of usage (it has been my food shopping bag for more than a year).

I have other two reusable bags made from recycled polyester (so recycled plastic bottles and plastic bags) that have seen probably 1800 uses each, maybe more, in their lifetime.

This material is probably a better choice for reusable shopping bags anyway; it can be made from plastic that can't be reused for food grade, it's incredibly tough, it folds down small etc.