Thanks for the answer. Let me start by saying that I appreciate you taking the time to answer this. I think the problem you touched upon is really important. One (unimportant) terminology issue: you refer to "standard" Q.M. as if it would exclude the measurement postulate. The "standard" Q.M. we learn (or some of us teach) at universities usually includes the measurement postulate.
If I understand correctly, you give two examples of classical systems in which a certain function (feedback, adaptation) canot be achieved say via a standard quantization procedure. But why is this surprising? Isn't it the case that, if I define my function or goal to measure with high precision both momentum and coordinate I can achieve this in classical physics and I cannot do it if the system is quantum? Or, if my goal is to copy some property (state) of a system into another, I can do it if that property is a classical one and I cannot do it if it is a quantum one (no-cloning theorem)?
The point that Leggett makes in this context (and which is correct, I believe) is rather of a different nature: he is speculating that, given zero decoherence, it might be the case that if the system is macroscopic/complex, Q.M. might fail to give an accurate description. For example, Q.M. (and this includes the measurement postulate) might predict interference and in the experiment we don't measure any. This would mean really that Q.M. is not a universal theory and it would be probably the greatest discovery in physics in decades! This, to me, is very different from the examples you give, which do not show that Q.M. is unable to describe a thermostat but that a certain functionality, formulated in classical terms, is not reproduced as such by a simple quantization procedure. (By Q.M. I mean here the Q.M.-as-usual, that is, including the measurement postulate.) In reality, both examples that you give are open systems: the superposition principle then cannot be applied as such, and decoherence has to be introduced in a careful way. But I don't see any reason why such systems cannot be described by Q.M. - again, including at a certain point the usual separation between the classical and quantum, and so on. I stress here that it is indeed annoying that we cannot write a description without these assumptions - but we already know that this is the unfortunate situation so far in physics!
Suppose for example that I have an ensemble of spins, from which I can extract spins one by one, pass them through a Stern-Gerlach apparatus, and determine the up or down value of each of them. This is a von Neumann mesurement. By extracting enough spins (but not all), I can calculate the temp. of the spin bath. Then I use this information to regulate the temp. of the bath in which the spins exist (they can be the spins in a solid, in which case I change the phonon temp., just by heating the solid). This in turn will regulate the temp. of the ensemble.
In the system above, there is no contradiction between Q.M. and the idea of a thermostat. You might object that, yes, but I have not given a complete Q.M. description of the computer used to calculate the temp. and so on. Indeed, but this has been the situation with Q.M. since Bohr's times. Such a unified description of the quantum and classical is lacking in every experiment done in quantum physics in the last 100 years, and there is nothing special with thermostats from this point of view.