My feeling is that for semi-simple g (at least classical g) this should be known to be true. But I cannot provide a reference now.
I am not sure about details, but the seemingly similar sounding conjecture sometimes associated with names of Mishenko-Fomenko A.S. Mishchenko and A.T. Fomenko (early 80-ies ?). ??). As far as I understand their conjecture is that any (not only semi-simple) g has maximal Poissin - commutative Poisson-commutative subalgebra in S(g). I am not sure did they conjecture that we can find int. sys. on any orbit. May be not explicitly. There are is a bulk of works on this conjecture. Some names - A. Bolsinov, Trofimov, Fomenko himself wrote books on it, many of his students worked on it. I would suggest to look at
The argument shift method and maximal commutative subalgebras of Poisson algebras
Dmitri I. Panyushev, Oksana S. Yakimova
As far as I understand they rise question not only for g, but also for any affine Poisson manifold (I am not sure this work or not - but Yakimova surely discussed it).
About quivers - Nekrasov's old paper contains some examples of int.sys. on quivers arxiv.org/abs/hep-th/9503157 . My feeling was that one can see (at least some of) quivers as moduli spaces on vector bundles on very degenerated curves and so these are in a sense Hitchin's system.