Most actually-existing books simply write that the student should be "mathematically mature", by which they mean that the student should know all of multivariable and vector calculus, most of differential equations, much of optimization, and some amount of analysis. And also be able to do proofs, and also have good intuition.

In short, most "applied" math textbooks aim themselves at Master's degree students or beginning PhD students in math itself, to teach material which can actually be taught to anyone with a bachelor's in science or engineering (except possibly computer scientists, whose continuous maths degrade because we never use them).

Ironically, the first math textbook I've ever picked up that didn't assume far too much background was my real analysis textbook, because analysis teachers assume that they are teaching the gateway course to "real math" and have to educate little babbies who only just got done with calculus.

If you want to learn something difficult that's above your level, and learning that something includes learning a prerequisite subject that's also above your level, you simply start at the node in the DAG that's at your level and work through the prerequisites if you're serious.

But yes, there is no substitute for learning all the things you need.

Exactly. And a surprisingly large number of books aren't well-written, exactly because they don't include that bit.

For most well-studied topics, finding the books that are good enough is not that difficult.