Could you cite the "much touted paper" you're talking about?
In my view, making the noisy physical magic states is the easy part of the distillation process. You reset a qubit, then rotate it 90° around the Y axis, then 45° around the Z axis. That's the magic state. Note that the tolerance on those rotations is forgiving: getting them to within 10°, 95% of the time, is sufficient. All the error correcting code stuff that follows has fidelity requirements an order of magnitude stricter.
As you note, there'd need to be some unforeseen obstacle for state prep to be the showstopper. Given how apparently easy it is to make these states, I think any obstacle like that would basically have to falsify quantum mechanics as we know it. It would be like finding out that light can't be diagonally polarized.
The ability to rotate 45 degrees is equivalent to the construction of a magic state, as you have correctly identified, as it is not a clifford operator, and adding it to your generators gives universal quantum computation on its own. You have pushed the problem sideways.
This is the paper, https://arxiv.org/abs/quant-ph/0403025, and it is well understood by the paper that the independence of the noisy magic states is necessary for the distillation process to proceed. Note that the probability of having some entanglement between your partial states goes up rather dramatically with the number of them that you have, and not obviously in a way that you can do anything about.
> Note that the probability of having some entanglement between your partial states goes up rather dramatically with the number of them that you have
Entanglement is not binary, it is continuous. If you start with states like CPHASE(5°)|TT>, a few rounds of distillation will have turned them into states like CPHASE(0.0000000000000001°)|TT>. Sure the output states are "still entangled", but the amount of entanglement is so negligible that you don't have to care. Such small distortions won't prevent trillion step computations from working.
In my view, making the noisy physical magic states is the easy part of the distillation process. You reset a qubit, then rotate it 90° around the Y axis, then 45° around the Z axis. That's the magic state. Note that the tolerance on those rotations is forgiving: getting them to within 10°, 95% of the time, is sufficient. All the error correcting code stuff that follows has fidelity requirements an order of magnitude stricter.
As you note, there'd need to be some unforeseen obstacle for state prep to be the showstopper. Given how apparently easy it is to make these states, I think any obstacle like that would basically have to falsify quantum mechanics as we know it. It would be like finding out that light can't be diagonally polarized.