The hydrogenoid atoms and ions, with a single electron, are the exception that proves the rule, because anything more complex cannot be computed accurately.

The spectrum of hydrogen (ignoring the fine structure) could be computed with the empirical rules of Rydberg before the existence of quantum physics. Quantum physics has just explained it in terms of simpler assumptions.

Quantum physics explains a great number of features of the atomic spectra, but it is unable to compute anything for complex atoms with an accuracy comparable with the experimental measurements.

The QED calculations with "14 digits" of precision are for things that are far simpler than atomic spectra, e.g. for the gyromagnetic ratio of the electron, and even for such things the computations are extremely difficult and error-prone.