No. The problem with making it "the responsibility of the supplier" is that we are both offloading responsibility and then unwilling to pay the price for the proper handling of the waste.
So basically along with really low prices, the supplier is bundling a free absolution of our guilts along with their pollution! Ain't that grand.
Yes, it should be the responsibility of the supplier. That's how every other balanced system works in economics. You only have issues when there is moral hazard when someone is allowed to shirk their responsibility. Trying to have someone downstream make up for it is a fool's errand.
In this case, the supplier should be required to put up collateral or purchase some kind of cleanup insurance for cleanup. i.e. there needs to be a basic regulation from the government on land granted to mining.