There are several issues.
The modern approach to QFT in CST considers the $*$-algebra of observables ${\cal A}$ as a fundamental notion. This approach permits, in particular, to define a theory in all globally hyperbolic spacetimes simultaneously (using a functorial description) and, from this perspective, to take advantage of locality and covariance in a very broad sense.
For instance, this approach leads to the noticeable result that the values of renormalization constants of a given theory (abstractly formulated) must be the same in every globally hyperbolic spacetime.
In this framework, states are algebraic: positive functionals $\omega : {\cal A}\to \mathbb{C}$. The the Hilbert space (von Neumann) perspective is recovered through the so called GNS construction. Given a state as before, it provides a Hilbert space, a representation of the algebra ${\cal A}$ in terms densely defined operators in the GNS Hilbert space, and a cyclic vector: the "vacuum" of the theory.
The double commutant of the concrete algebra of operators in the GNS Hilbert space, in principle, gives rise to a von Neumann algebra of observables in that Hilbert space.
An immediate problem arises from the fact that this new structure is not invariant under the $C^*$ or $*$-algebra morphisms relevant at the level of abstract algebra ${\cal A}$.
That is because this structure is connected to the uniform norm (the $C^*$ case) or it is not topological ($*$ algebra case), whereas von Neumann algebras are topological objects fundamentally related to the weak and strong operator topologies, which are weaker than the uniform one.
For instance, suppose that an abstract observable (an element $a$ of ${\cal A}$) is represented by a selfadjoint operator $A$ in the GNS Hilbert space associated to an algebraic state. $A$ is a very complex object, in particular it has an associated spectral measure made of orthogonal projectors with an operationistic meaning (the "elementary propositions" which refer to $A$).
There is no way to see this associated structure as a part of ${\cal A}$. In fact, a spectral measure is a topological structure connected with the strong operator topology of the Hilbert space (the spectral projectors belong to the von Neumann algebra generated by $A$) and there is no chance to lift it to the more abstract level of ${\cal A}$.
In this sense a von Neumann algebra is too rigid to be used in algebraic QFT in curved spacetime if one want to deal with the power of the locally covariant formulation. It may make sense However, once a state has been chosen and in a specific spacetime.
