Quantum Mechanics is actually incredibly conservative, with theoreticians simply refusing to abandon notions shown to be incompatible with reality for over a century now.
Same with Spin. Okay, so particles aren't literally spinning spheres. We get it. Stop pretending they are and then Spin won't be so mysterious any more!
It’s just an evocative name?
Okay then, if it's impossible, don't print what it isn't. Put down in writing what it is. That is all.
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”.
Note that MWI as originally formulated by Everett does not work this way. Eg quoting from the Discussion section of his thesis:
> [I]t seems to us to be much easier to understand particle aspects from a wave picture (concentrated wave packets) than it is to understand wave aspects (diffraction, interference, etc.) from a particle picture.
or
> This view [ie Everett's view, as described in the thesis] also corresponds most closely with that held by Schrödinger.
The main point of MWI is not to go back to a more classical picture of point particles, but to remove wave-function collapse and Heisenberg cut from the description.