You can develop a similar plan for teaching programming that starts with basics and progressing (variables, linear control flow (aka, no conditionals or loops), conditionals, loops, functions/subroutines, complex types like records, recursion, modules and classes; in roughly that order). You aren't hiding anything, you aren't misstating anything, you just don't move on to advanced things until it's time to. Of course, this progression would probably be a lot faster than the arithmetic -> calculus progression.

The idea that there are "basics" that can be safely ignored once learned is patently false. Everything I've read from great performers in any field- piano, cooking, mathematics, physics, etc- contains constant encouragement to return to and refine one's understanding of the fundamental skills of the practice. I did indeed spend a long time practicing addition before anyone informed me that it was time to "move on" to multiplication, and it wasn't until I graduated college with a degree in aerospace engineering that I had read a few books about the history of mathematics and found out that the manipulation of Arabic numerals via "addition" or "multiplication" was simply shorthand for counting, which itself is a fundamental mathematical art. What was taught to me as "multiplication" is actually "one particular algorithmic notation for representing the counting of multiple sets", not "objective mathematical law".

Had I disabused myself of the notion that there was any part of mathematics that I could "stop thinking about" because they were "little", I would have made a much better showing in school.