It's not pushed to the producer, it's pushed to the consumer, just as charging more for trash does (which may be ideal depending on who you ask). Regardless, all this does is keeps those who have less from obtaining more, others will eat the costs. Seems everything that is disliked these days always has government taking money as the suggested solution. Even if it was the only way, plenty is already taken to rationally use it to help without using the tool punitively.
What cost isn't eventually pushed to the consumer?
You pay more upfront, you pay more for disposal, you pay more in tax so the government can handle it, or you pay with a degrading environment in which to live.
Forcing producers to account for the full life cycle internalizes that particular externality and makes it an area they can compete in.
This reduces the situation down to simple zero sum as if public spending waste and several other factors don't exist. It doesn't force producers to do anything. Often these types of punitive taxes just feed government coffers and don't affect the problem to the contrary of what idealists told them with simple napkin math when proposed. Encouraging better handling of these things can happen via many awareness and grant initiatives with the vast amounts of monies already taken. You don't have to take more.
Taxes are pretty good at raising awareness, since they're quite controversial. Of course, that increase in awareness isn't necessarily going to result in better actions--there are likely people who will deliberately sabotage such efforts in protest.
If the only solution here is to consume less, taxes could work for the majority of the population. If the cost of consumption goes up, actual consumption goes down. But a significant minority of the population would probably be willing to eat the extra cost, which is unfortunate.
Not all waste producers are individuals, either. Industrial processes often produce a lot of waste, and the products at the end of the chain don't always go to individuals; they might go to the military, for example. Big companies are more likely to have the money and lawyers to evade taxes and fines, which impedes regulation and rewards companies that don't play fair. These issues aren't seen by consumers, so we can't vote with our wallets, either.
Raising awareness via traditional methods seems like a good long-term strategy, but what about all the people who simply don't care?
Ultimately, we're going to need to tackle this problem from multiple angles. It's unlikely that a single tactic will yield satisfactory results. Regulation and taxes are probably going to need to be part of our plan, but they won't be sufficient on their own.
