Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces

We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak weak approximation if $k$ is a number field and the Hilbert property if $k$ is Hilbertian of characteristic zero, provided that they contain a $k$-rational point lying neither on any $4$ of the $56$ exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado--Testa--V\'arilly-Alvarado, and Festi--van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration.