Interesting, could you provide a reference for $PH^{\#P}$ and the problem of its collapse, please?

It does not. The answer you quoted says that the collapse of CH is closely related to another problem with constant-depth threshold circuits. But it does not clarify the relation. According to the complexity zoo $TC^0=CN^1$ imples $CH=PSPACE$ which in turn implies the collapse of $CH$. Thus, constant-depth threshold circuits provide a sufficient condition for the collapse of $CH$, not a consequence.

@Huck Bennett - I don't understand your criticism: computing functions is usual in everyday math, and indeed your laptop not only decides problems but does also compute functions. Considering only decision problems limits our view. On the other hand, if men had had considered only simplest things, we wouldn't have had computers and not even wheels.

@Huck Bennett - Do we need motivations for such a natural generalisation of function problem classes? A function problem is a very natural and well-known notion in computational complexity. It is thus very straightforward to ask what are the function problems solved by probabilistic or quantum Turing machines. If we ask similar questions for decision problems, why not for function problems?