I think my own education followed the “traditional” way of learning computer science, albeit after years of tinkering with computers and programming as a kid. But I often felt that some concepts were much simpler when you made the connection with something concrete.
Of course, you would not use analogy for formal proofs, but it helped build a mental model incrementally instead of having to conjure various concepts out of thin air and draw connections between them.
The key part was studying enough mathematics. When your homework involves several proofs by induction every week, you get used to a certain way of thinking. If someone then shows you a dynamic programming algorithm, you will probably notice that the same process you are using all the time can also be used for designing algorithms.
You were doing the hard work in the mathematics classes, while dynamic programming was a simple topic you could understand after a single lecture. Definitely simpler than balanced search trees. But once the average CS student was no longer comfortable with mathematical proofs, they had to do the hard work while they were learning dynamic programming.
Of course, you would not use analogy for formal proofs, but it helped build a mental model incrementally instead of having to conjure various concepts out of thin air and draw connections between them.