I have a little collection of slide rules. I love those things.
I'm not old enough to have used them to do calculations, but I find them extremely useful to explain logarithms and how multiplication can be represented by the sum of logarithms. I actually work with grad students who should know these things, but watching it in a slide rule on their hands really helps to build intuition.
AFAIU the problem, from a theoretical point of view, is not that you need to pass the self function, but how to define it. In pure lambda calculus, there are no named functions, just lambdas, so you actually need the Y combinator for recursion.