The short answer: No, classical mechanics is not recovered in the $\hbar \rightarrow 0$ limit of quantum mechanics.
The paper What is the limit $\hbar \rightarrow 0$ of quantum theory? (Accepted for publication in the American Journal of Physics) found that
Our final result is then that NM cannot be obtained from QT, at least by means of a mathematical limiting process $\hbar \rightarrow 0$ [...] we have mathematically shown that Eq. (2) does not follow from Eq. (1).
"NM" means Newtonian Mechanics and "QT" quantum theory. Their "Eq. (1)" is Schrödinger equation and "Eq. (2)" are Hamilton equations. Page 9 of this more recent article precisely deals with the question of why no wavefunction in the Hilbert space can give a classical delta function probability.