The Bohr model is incomplete and has drawbacks. But one thing is a mystery to me. Why did it so successfully calculate the rydberg series with quite good number of correct digits?

Having such a good prediction one would expect that ther exists an extension or modificaiton to it, although we may not have found it, that would yield the correct model. But today we have abondon it completely and use QED. I would expect that from QED we should be able to derive the math in the Bohr model, my subquestion is if there is such a derivation and it would also be super duper if we could sketch it here.