Literally every QM textbook ever printed starts the explanation with: "Spin is like a rotation but it isn't really because that's impossible for an electron if it were a tiny spinning sphere."

Okay then, if it's impossible, don't print what it isn't. Put down in writing what it is. That is all.

Spin is, aiui, related to an algebra that satisfies the same relations as the operators for the usual sense of angular momentum.

However, unlike the operators for the usual sense of angular momentum, there are additional solutions that appear, which have the half-integer spins.

And, in a sense the spin contributes to the angular momentum.

I don’t see a problem with saying “it is in some ways kind of like if it were spinning, but it isn’t quite the same. Here is the math to describe it, and how that math is like and how it is unlike something spinning.” . It seems better than just saying “here is the math which describes it”.