And that is exactly why it isn't a paradox in quantum mechanics. Quantum mechanics is consistent for the domain of questions it was designed to ask: particle scattering experiments. It's when people push it into philosophy or other unintended domains that they run into problems.
Quantum mechanics is also totally consistent for materials science problems, astrophysical problems (pending gravity), chemical problems (including human beings), and so on. It's really the philosophy that has a quantum mechanics problem. ;)
A paradox is an internal inconsistency. Noone claims that quantum mechanics is the fundamental theory of everything. Like the convergence of a taylor series, there is a range of energies within which it produces correct calculations, and if you go outside that the result is undefined.
Strictly speaking that is a QFT problem, or really a standard model problem, moreso than a quantum mechanics problem. In order for it to be a quantum mechanics problem it would have to happen everywhere in quantum mechanics, and the single electron Schrodinger equation hydrogen atom definitely has no (theoretical) problem.
