> Shor's algorithm requires not only the logical qubit to be stable (the superposition of 1 and 0, but also the phase [..] to be noise free to an extraordinary degree. There are rotations in very small angles many, many times in the computation.
This is wrong. Those rotations are not the hard part.
When Shor's algorithm is compiled using qubit recycling [0], factoring an n bit number uses 2n rotations that aren't a multiple of 45 degrees. For relevant number-to-factor sizes like 2048 bits, performing these rotations to a precision of ±0.0000000001° is more than sufficient. Rotations smaller than the required precision can simply be skipped. If you're asked to rotate by 0.00000000000000000001° ± 0.0000000001°, not rotating at all is within tolerance.
±0.0000000001° is more precise than would be reasonable with physical qubits, but it's achievable with logical qubits that can do 45 degree rotations. You decompose the arbitrary rotation into a sequence of a ~hundred 45 degree rotations that approximate the desired rotation [1].
It's also possible to replace all rotations by additions into a special re-usable "phase gradient state" [2][3]. When factoring a 2048 bit number, preparing a phase gradient state of sufficient quality only requires around a hundred non-multiple-of-45-degree-with-precision-±0.0000000001° rotations, instead of thousands.
If you count it all up, the precise rotations account for maybe a hundred thousand error corrected gates. The algorithm as a whole uses a billion to a trillion error corrected gates, depending on exactly which gates you count. The precise rotations are a negligible part of the difficulty; they make up less than 0.01% of the required work. From the perspective of superconducting qubits, the hard part is actually just idling a surface code qubit! All the operations are implemented using variations on idling.
0: https://arxiv.org/abs/quant-ph/0001066
1: https://www.mathstat.dal.ca/~selinger/newsynth/
2: https://arxiv.org/abs/1709.06648
3: https://arxiv.org/abs/1803.04933