That probably doesn't help. A cotton bag requires about 130x more resources to produce than a standard HDPE carrier bag. If you re-use a plastic bag three times, you'd need to use your cotton bag nearly 400 times to break even. The cotton bag is likely to wear out substantially sooner than that. The plastic bag can be disposed of responsibly by recycling, incineration or as a container for non-recyclable domestic waste.
https://assets.publishing.service.gov.uk/government/uploads/...
It doesn't really matter who you blame in this case, because the solution is the same either way: stop allowing stores to offer free plastic bags.
And if anyone would accuse my of rationalizing my inefficiencies, then that is just one more reason why you cannot trust individuals to make the right choices.
India recently stopped plastic bags in stores, and most shops don't have bags that you can buy (plastic or otherwise). You have to bring your own bags, or you can carry your stuff in your hands :P It is so painful that people don't forget bags while grocery shopping.
I don't have an answer to this specific scenario that you described. That said, I wasn't saying we should all suddenly become 100% socially responsible overnight - it is just not practical.
There are lots of things that are super easy to do, and yet have positive influence on the environment, society etc. We can keep a handful of cloth bags handy, when we go grocery shopping. This isn't hard to do, and will help reduce plastic bags. We don't have to wait for legislation to act on this. We don't have to replace our perfectly working phone, every other year (or every year) and throw it in the landfill. And so on.
These are definitely small compared to the pollution caused by corporations. But corporations exist to sell to consumers. Plus, change has to start somewhere. Why not start at home?