Earlier quoted context omitted.
Littering/dumping is a much different issue than disposing of your waste through the wrong channel. The problem you are describing isn't because people use the garbage instead of the recycling.
If we only had paper bags and such, littering wouldn't be a problem. Single use plastics of any kind need to be banned universally. Pseudo multiple use plastic such as bags needs to be taxed like alcohol and tobacco, through the roof, until alternatives become economically viable.
I would be more amenable to the idea if they made a bag that does not get torn (e.g. armor the paper with cotton or bamboo thread), and is leak-resistant (e.g. waxed), but that somehow takes the back seat.