As someone who’s worked for the CMS experiment of LHC, I can assure you QED is fucking well tested in reality and isn’t “based on a single experimental value”, so the paper is indeed nonsense.
It isn’t the answer to everything because the electromagnetic force isn’t the only fundamental interaction and we don’t have a theory of everything. But QED explains one of the fundamental interactions far better than the paper tries to paint it. See The Relativity of Wrong by Isaac Asimov.
I think the linked article agrees with you that QED is well tested.
Their point (IMO) is that there is one free parameter in QED that was not highlighted as such. There is nothing wrong in free parameters, we know very little of the universe. As far I know there are free parameters in the standard model, but it is clearly stated as such so there is no problem.
The electron's magnetic moment is not a free parameter that must be determined experimentally. It can be computed from other parameters, as done for example here: https://arxiv.org/abs/hep-ph/0406325.
