Why a MOSFET in accumulation mode is not conducting?
In fact, when an accumulation region is formed under the channel, this layer of charge can conduct current. In the past people have built accumulation mode field-effect transistors (see here: http://ieeexplore.ieee.org/document/192780/)

In a conventional MOSFET, however, the reason why the channel is not conducting is that the accumulated in the channel is of the opposite type as that of the source and drain. For instance, in an enhancement-type N-MOSFET in cutoff mode of operation, the accumulation region would be p-type and the source and drain are n-type. Electrons would not be able to flow across the p-type body due to the barrier between seen from the source to channel. In this case, applying a positive voltage on the gate to form an inversion layer under the channel is required to lower the source-side barrier, allowing electrons to flow from source to drain. 

Here is an example of the gate voltage lowering the source-side barrier for electron flow.