I think it would be more pragmatic to have environmentally friendly single use bags available for a fee rather than wasting all those "reusable" bags.
I admit I am terrible at it. If I ride my bike then I bring something the carry the stuff home but if I take my car, even though I put bags in my trunk, it never even crosses my mind to use them :( Maybe posting this comment will help me remember next time.
On the other hand, my apartment complex demands we put our trash in plastic bags so I use the bags I get at the store for that ... sigh ...
I actually worked in a hardware store for a while, and after the old supply of plastic bags ran out it was common practice for customers to either bring their own bags, carry stuff out in their hands, or use one of the hundreds of small cardboard boxes or trays that were set aside on receiving day when the shelves were stocked.
I see it most days at the supermarket near my local train station: people get off the train and stop by to buy stuff. I see a good number of people buying "reusable" bags because they don't have any bags with them... it is ridiculous not to be able to have access to single use bags in this scenario. Mandate that they be paper or compostable if needed but "no it's banned" is not pragmatic at all.