No you are correct.
There is at this point like 40 years of quantum chaos research. I dipped my toes into it a bit in grad school. In the eighties it was believed that chaos might be the key to understanding the quantum-classical transition. It's not a very cool area of research anymore, mainly because it hasn't really led anywhere.
We know that when applying decoherence to quantum mechanical systems we recover the original classical (and potentially chaotic) equations of motion. This basically "solves" the chaos problem but doesn't actually help with measurement, because it doesn't cause collapse of the classical probabilities. This is what is referred to as "superselection".
In terms of cat analogies, the quantum wavefunction is a superposition of amplitudes a|dead> + b|alive> and these amplitudes cannot be chaotic because the Schrödinger equation is linear. When the system becomes decoherent, it becomes a classical probability distribution p(alive) that evolves under potentially chaotic classical motion. However it's still probabilistic, the cat remains neither alive nor dead. We haven't "measured" anything.
The confusion (and this is what Schlosshauser talks about) is that for some people (famously Ballentine) who follows the ensemble theory of quantum mechanics, there is no meadurement problem at all once we have a classical probability distribution. This view is not very popular however.
So anyway the point is that the existence of classical chaos can be straightforwardly explained by decoherence without any collapse at all.