They worked pretty well when I lived in MI (although non-deposit containers would still become litter). Now I live in TN, with no deposit, and cans definitely get recycled less, even in relatively liberal parts of Nashville.

https://www.infographicsarchive.com/which-states-recycling-t... (which in turn cites https://www.ball.com/sustainability/real-circularity/50-stat... ) seems to show deposits correlating with better recycling rates. MI got a poor score overall, but in cans it's on top.

89% of deposit bottles are returned and recycled in Michigan