As a kid growing up in a small town in the Midwest in the 1960s and early 70s, I certainly remember layaway... and Christmas Club accounts.
(Christmas Club: A savings account where you put $XX dollars away each week, with the idea that, in time for Christmas, you'd have enough squirrelled away to do your shopping.)
Both ideas make a lot of sense. They are both about financial responsibility, about setting limits/boundaries, about living within your means. Nowadays with credit cards and endless advertising and stores being stocked and decorated for Christmas right after Labor Day, it's even more of a challenge to stay within one's means. Christmas Club and layaway may be old-school, but sometimes, old is actually new again.
PS One other thing about layaway: It was a great way to "hide" gifts from the kids. Kids can't find the toys if they're sitting at Kmart or Walmart. ;-)