What problems? Note that Aaronson, in the article (as well as a number of commenters in the comments) gives decoherence as precisely the answer to the issue you raise.
Another is that decoherence still does not explain the origin of probabilities. I.e. there are models that shows how a pure classical system may arise. But we still have no explanation how a classical device interacting with a quantum system gets the value it measures.
consilient answered this one.
> decoherence still does not explain the origin of probabilities
The amplitudes corresponding to each decoherent branch are the right ones to give the Born rule probabilities. Which, given that the Born rule is empirically observed to work, is probably as good as we're going to get unless (as I doubt will ever happen) someone manages to find a way to derive the Born rule from unitary dynamics.
Outside the perturbative regime, there isn't a preferred basis because there aren't any classical observers.