Why is reducibility an adequate measure of difficulty?
It's not; it provides a relative measure of difficulty, relative to the kind of reduction under consideration.
To me, the most natural measure of difficulty is worst-case asymptotic complexity.
That's the idea, but for NP-complete problems in particular we do not know their (worst-case asymptotic time) complexity. By introducing a relative measure of hardness, we can say "A is at least as hard as B" without knowing how hard either of them really is.
having an algorithm to solve A is not at all useful for solving B, and so neither problem is reducible into the other.
That's not the idea behind Karp reductions which we use to define NP-hardness. Read up on the definition.
this complexity is "the worst" possible in NP
There is no such thing as a worst possible complexity; all polynomial complexities are allowed, and there are problems of arbitrary polynomial complexity. You find proofs for this fact in textbooks on the matter.
It seems at least plausible to me that there might exist two NP problems A and B which both have the same worst-case asymptotic complexity, and that this complexity is "the worst" possible in NP, but the problems are so different that having an algorithm to solve A is not at all useful for solving B, and so neither problem is reducible into the other.
Keep in mind that complexity classes like P and NP are rough; they ignore polynomial factors. So say we have $\Theta(n^k)$ algorithms for NP-complete $A$ and $B$. We know that $A \leq_p B$ and vice versa, but all that tells us that we can use the algorithm for one to solve the other with polynomial overhead. The degree of this overhead is not specified, and may very well be in $\omega(n^k)$.
Bottom line, you have not yet understood/absorbed the basics and struggle with your faulty intuition contradicting statements you find. I recommend you check out our reference material which covers a wide array of basics and common misconceptions.
Keep in mind that the theory around P and NP defines a measure of hardness. There are certainly others, and some are more appropriate for making some decisions than others.