It'd be nice if we could find a way to extend that type of system to other types of waste, but that gets more difficult.
No, they really don't. Having spent enough time in states that do and states that don't, I can tell you that having a deposit system encourages some bad behavior and doesn't really solve the problem.
1. It encourages scavenger people that dig through recycling and garbage making a mess of things in order to find large bottles with a deposit. On face value you think, 'yay, we get free sanitation workers cleaning up the city'. The reality is, they find the most efficient way to dig through and don't care about throwing disposable trash all over to get the loot.
2. Hoarding. I've seen people's kitchens and basements filled with cans because they want to do it all in one shot or for some special occasion. It makes a mess and is super annoying. You like fruit flies? This is how you get fruit flies.
3. It's inefficient. People who have the means will still not care about collecting the deposit because it's a PITA to go down to the store with the empties. The only people who get excited about it are the same people who think a tax refund is actually a gift from the government..
While it might solve some of the above problems, you won't find much luck making deposits $1/bottle or something in the US because it's only going to make life more of a pain for the poor. It's not a workable solution.
A few years ago the government suggested taxing plastic bags to gain revenue, but it was discarded since the environmental organizations concluded it would be bad for the environment (as more people would throw garbage directly into thrash cans/buy thrash bags).