Doesn't matter. If it's your responsibility, you pay. If it's the automaker's responsibility, the automaker pays, and then carries that cost forward into the price of the cars, so you pay.
Whose primary responsibility for making it happen? I mean, only the manufacturers can make it happen [redesigning batteries to be easily recyclable], but they won't, so someone has to make them, but customers won't, so the government will have to. This is one of those things the government is for, like banning trans fats and ozone-depleting refrigerant. Making all batteries more recyclable should be top of the list for this decade.