Around 1927, there was a transition from the "old" quantum mechanics to the "new" quantum mechanics. There were lots of problems with the old quantum mechanics, and the shortcomings of the Bohr model were just one such problem. In the new quantum mechanics, the idea of classical trajectories for particles is completely abandoned. They propagate as waves instead.
Ultimately, the reason for preferring the new quantum mechanics is that it is in better agreement with experiment. One decisive experiment was the 1926 Bothe-Geiger experiment, which disproved Bohr's idea that light could still be described classically, and only the atom needed to be quantized. The old quantum mechanics also got various things wrong in its description of the atom, e.g., it predicted the hydrogen atom to have an orbital angular momentum of $1\hbar$ in its ground state, rather than the correct value of zero.
So in summary, Bohr's third postulate was not one isolated problem with the old quantum mechanics that needed to be fixed for one specific reason.