if we accept Fourier Analysis as correct

I'm not very sure what Fourier Analysis is. If it is about Fourier Transform, yes, the FT is correct in a mathematical way. In physics, however, it is not that easy to say when FT results make sense or not. In order for FT to work, it needs the concept of infinity. In the real universe, however, everything seem to be bounded in a way or another. Infinite is not real.

Even if it were for infinite to be real and applicable in physics (real as in heaving examples in the physical universe, examples independent of our own thought and imagination), it still remains the problem of mapping from physics to math, and back to physics after some math processing. That is why we always have to check if math results make sense, then have to validate them experimentally. Not saying the encoding/decoding between physics and math was incorrect, saying only that it is tricky, and that we do not have any known tool to prove the encoding/decoding to/from math was made correctly, other then double checking the results experimentally.

There are more traps with mathematics. It doesn't have any axioms related to causality, or to time. Causality and time are present everywhere in physics, but in math they are absent. These two treats, causality and time, are left to be represented by our skills in encoding/decoding a problem between the physics domain and the math domain. Very prone to error.

Mathematicians have no problem operating with infinity, having some infinities bigger than others, and so on. Math is all about building an entire world on a set of axioms, but the axioms can be changed. In contrast with math axioms, the physics laws can not be changed, or selected upon our wish.

At first, the math axioms were deduce from the physical observations alone. Then the axioms were modified, or extended, giving birth to different worlds, some results were contradictory (e.g. sum of angles in a triangle is not mandatory 180), but they were coming from different sets of axioms, so no problem, mathematically it is correct. With time, the set of axioms were changed, either to cover more of the physical world, or to allow more advance in math. In time, math became a world in itself. At first, all the math results were about the physicality of our world, but not any more. 1:1 mapping between math and physics was long ago. Now, a math result may or may not be applicable back to physics.

Math is great when applied with care, and when the conclusions are validated experimentally. But math is not evidence. My point is, the FT can work great and be correct mathematically, yet entanglement can no more than synchronized waves.

More arguments for entanglement as being synchronized waves, entanglement does not lasts forever, as suggested in all popular science. That would be only in an ideal situation. In practice the decoherence happens rather fast, just like two oscillators that were once synchronized then separated and let to run freely. This is the main reason why quantum computers in practice only have a handful of qubits. The wave synchronization (of whatever quantum object is used as a qubit) is perturbed, the fancy word is decoherence, or disentangling. This is no different than oscillators getting out of sync. Yes, they qubits are really small and easy to be perturbed and put out of sync.

Another hint that entanglement has nothing weird, is that entanglement happens only when the particles are put in close vicinity (or even born together). It is known that oscillators that are close enough that they can interact, tend to synchronize, and get in lock with each other. It happens at macroscopic scale, too. In EE for example, if you try to do intermodulation distortions, you might sometimes need an isolator between the two generators when summing their waveforms. If they can exchange energy and if they are of a close frequency, the two instruments will tend to get in sync and lock to a single frequency (which is unwanted in this case).

I do not know why the spin pairs, best guess is that is because that's the state with minimum energy, thus the most stable and the most probable to be encountered, but I don't have an understanding of spin (understanding as in growing an intuition about it).

The puzzling property to me is not entanglement, but quantization. Why something that is very small can only exist as a certain "chunk", called quanta? This is a property that seems to be present in everything from the physical world, as long as it is very small. I suspect it is related with the fact that infinity can not exist in the physical world.

Side note, similar with infinite, the concept of "nothing" is also not real, as in not present in the physical world. Just like the infinite, nothingness is an imaginary construct. Nothingness can not exist in the physical world.

Talking about nothingness being only an imaginary construct and not real, because it is related with infinite being imaginary, and because it was asked in this thread a page ago:

What is nothing?