If you want to measure spin in the x direction, you need to know what "the x direction" is.
Moreover, in a vector representation of particle spin, you take the two directions "up" and "down" (which are 180deg apart in Euclidean space) to a vector basis ("90deg" apart or having a zero inner product).
In the case of measurement, that's a basis too. Without vector subspaces of the Euclidean space you do measurements in, how do you measure anything?
Although you might believe you don't need a coordinate system, you will find it difficult I imagine to do any science without one or two.
Or mathematics, like basic addition, say.
A physicist might argue against your POV by saying you can't really avoid vectors and coordinate systems, they're just too ubiquitous. A mathematician might argue that vector spaces are also ubiquitous mathematical structures, that they happen to explain a lot of physics isn't the point (unless you're a physicist, see previous sentence).
I don't understand what you mean by "solve the double slit problem".
Can you solve the delayed-erasure experiment problem the same way?
It was also ubiquitous for Minkowski to set time ITSELF, an INSTANT of it, proportional to the speed of light. Wrong. Space has no inertia, and if you expect the math will still work by endowing space with inertia (by giving it a coordinate system that has physical meaning in terms of distances and velocities), that would be just as wrong. This is the proportional math equivalent of dividing by zero. It matters little whether after that, you decide the quantity needs to be complex so as to give it a preferred direction (no reverese time travel), or to start doing Pythagorean geometry with it because you know how to build that geometry into transformational boost matrices. It was wrong before boost matrices were even invented.
Look, trying to understand entanglement with vector math is going to be about as productive as working out the Periodic Table using billiard ball collisions as your only modeling tool.
There was a time, on these forums, that I still believed you could pack up the geometry of Special Relativity into a tool kit and use it to go exploring the spin energy dynamics inside of electrons or quarks. I now realize, this was a mistake. Entanglement is what makes it a mistake. Entanglement is what makes bound energy possible. For that matter, it is also what makes unbound energy propagating at c possible. It is not possible to attach (with inertia) a photon, or the path of a photon to inertialess space, because absolute position does not exist. Absolute veloctity, even an invariant one like c, does not work the way you think it does either, and for the same reason absolute position does not.
Inertia is not a property of space. It is a property of energy. And neither energy nor inertia exists without a deeper understanding of an instant of time, and entanglement.
Absolute time, however, is another matter. It is because the quantum field that pervades all of what we call "space" has no quantum spin (spin=0) that entanglement has meaning, and in particular, that this allows particles with spin both to exist and to PERSIST indefinitely as a function of the passage of time at various rates associated with relative motion and other factors. An absolute instant of time spanning the known universe is not possible with Minkowski's version of time, which would be limited to a consideration of unbound energy propagating at c, (the only state in which it can persist in unbound form), or of the centers of bound energy (particles) at velocities <c. He covered nearly every contingency in a universe of events, but did so in a manner that for over 100 years has obscured understanding of possibly the most important energy transfer event in this universe.