"Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric." this not correct. Solutions should transform as a representation of the rotation group. They need to be covariant not necessarily invariant. Only a full shell is invariant.

" there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry" there are indeed infinitely many, but only $2(2{\cal l} +1)$ are linearly independent. You are free to choose which ones by choosing the so called quantization axis.

Once the set of linearly independent orbitals is selected, they can be filled following the exclusion principle.

"Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric." this not correct. Solutions should transform as a representation of the rotation group. They need to be covariant not necessarily invariant. Only a full shell is invariant.

" there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry" there are indeed infinitely many, but only $2(2{\cal l} +1)$ are linearly independent. You are free to choose which ones by choosing the so called quantization axis.

Once the set of linearly independent orbitals is selected, they can be filled following the exclusion principle.

Note that an accurate calculation of orbitals requires so called self consistent field theory.

"Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric." this not correct. Solutions should transform as a representation of the rotation group. They need to be covariant not necessarily invariant. Only a full shell is invariant.

" there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry" there are indeed infinitely many, but only $2(2\cal l+1)$ are independent. You are free to choose which ones, which is why you can choose the so called quantization axis.

"Really, the set of solutions to the angular equation should be closed under rotation, since the equation itself is also rotationally symmetric." this not correct. Solutions should transform as a representation of the rotation group. They need to be covariant not necessarily invariant. Only a full shell is invariant.

" there ought to be infinitely many possible states for an electron in a given shell, due to the continuous rotational symmetry" there are indeed infinitely many, but only $2(2{\cal l} +1)$ are linearly independent. You are free to choose which ones by choosing the so called quantization axis.

Once the set of linearly independent orbitals is selected, they can be filled following the exclusion principle.