This isn't just folk wisdom either. There's an active subfield of cognitive science that focuses on working memory/cognitive load and math instruction. Broadly speaking, it's hard for students to successfully transition to even basic algebra if they don't have basic calculations down.
If you don't get the basics of arithmetic it is incredibly hard to grasp the idea of "equals" as a pivot point. Division by hand helps to give us a hint that numbers can be broken up into constituents and still be the same.
You internalise a whole set of behaviours, typically via repetition, and it unlocks understanding beyond itself imo
It's all about building higher and higher abstractions. Arithmetic - > Algebra -> higher order algebraic algorithms.
For example, when I taught differential equations, my non-scientific observation was that otherwise smart students struggled more with getting the algorithms down, if they struggled with algebra. Having to constantly jump down to a lower level of abstraction while performing the algorithmic steps is what caused confusion to mistakes.
