Applying the principle of Occam's Razor to Quantum Mechanics
No. If the classical path was assumed to be the only path, there would be no quantum theory. It would just be classical. And clearly from the need for and success of a quantum theory that explains things outside the domain of the classical one, we know the world to be following quantum rules.
In Feynman’s highly readable QED he shows that assuming only the classical path fails to explain reflection from a glass slab. Experimentally the reflection depends on the thickness of the slab and he shows how it can be explained by the “all path” approach.
One needs to be aware of when it makes sense to use Occam’s razor. We can’t rule out a successful theory with a less successful one only because the less successful one is simpler. It must be used when choosing between things that have the same domain of validity. For instance, “the particle takes all paths” vs “the particle takes all paths and god exists.” Here both theories make the same testable predictions but one has an extra untestable factor. Occam’s razor says pick the simpler one.
First of all, It's not true that the weight of the classical path is the highest in Feynman propagator. It's the one whose contribution doesn't get cancelled out by other paths in the limit when the action is very large compared to $\hbar$.
In all other cases, the non-classical paths play a crucial role in the results of an experiment. Quantum mechanical predictions which deviate from classical predictions are absolutely measurable and are measured all the time. For example, see the double-slit experiment. Moreover, even if quantum mechanics were to adjust classical mechanics by just a little bit, it couldn't have been avoided using Occam's razor because it would still provide us newly accurate information that is not available from classical mechanics. Also, there are other fundamental reasons as well as to why quantum mechanics is unavoidable. For example, you cannot explain the stability of the atom in classical mechanics. Since quantum mechanics does explain this--something that classical mechanics cannot, Occam's razor doesn't rule out quantum mechanics at all.
Finally, even barring experimental inadaquecies of classical mechanics, there is no clear way to assert as to whether quantum mechanics requires fewer assumptions or classical mechanics. If anything, since classical mechanics is arrived at after assuming the criteria that make a quantum system approach its classical limit, one can argue that quantum mechanics requires fewer assumptions.
Besides, the language used in the paragraph you quote is highly misleading. It's not true that you can't know the initial state of a quantum system precisely. You can absolutely know that. For example, the state of a spin half particle is precisely spin up in a certain direction if I measure it to be spin up in that direction (and I can do that). The story about knowing either the precise momenta or positions is a bit different because there is simply no physical state which has a definite position or momentum, however you can still specify the initial state (or any subsequent state) of a quantum particle perfectly precisely either by specifying its wavefunction in a certain basis (which you can do) or by specifying its quantum numbers with respect to some appropriate complete set of commuting operators (which is also something you can do).
First of all, it is not true that the maximum of the probability always lies near the classical path - two-slit experiment or discrete energy levels are proves of that. When it is indeed the case, we call it quasi-classical approximation.
Secondly, even when the maximum of the probability is close to the classical path, we are still interested in more precise calculation - the above mentioned quasi-classical approximation is a method for calculating quantum corrections to classical behavior.
Finally, Occam's razor is an empirical principle, which is grounded in our intuition and on occasion in arguments following from the probability and information theory. Occam's razor is in no way a substitute for or a counter-argument against the experimentally verified laws of physics.