  Adsorbate engineering offers a seemingly simple approach to tailor spin-orbit
interactions in atomically thin materials and thus to unlock the much
sought-after topological insulating phases in two dimensions. However, the
observation of an Anderson topological transition induced by heavy adatoms has
proved extremely challenging despite substantial experimental efforts. Here, we
present a multi-scale approach combining advanced first-principles methods and
accurate single-electron descriptions of adatom-host interactions using
graphene as a prototypical system. Our study reveals a surprisingly complex
structure in the interactions mediated by random adatoms, including hitherto
neglected hopping processes leading to strong valley mixing. We argue that the
unexpected intervalley scattering strongly impacts the ground state at low
adatom coverage, which would provide a compelling explanation for the absence
of a topological gap in recent experimental reports. Our conjecture is
confirmed by real-space Chern number calculations and large-scale quantum
transport simulations in disordered samples. This resolves an important
controversy and suggests that a detectable topological gap can be achieved by
engineering the spatial range of spin-orbit interactions.
