It seems that such a measure for mixed states is fundamentally impossible, since you can have both entangled and separable states which have exactly the local expectation values. For pure states, monogamy of entanglement ensures that the impurity of a reduced density matrix (which can be infered from the expectation values of local Pauli operators) is directly related to entanglement. However for mixed states, this is not the case, as the following example will homefully make clear:
Consider a two qubit system, in which the two reduced density matrices are maximally mixed. In this case, it is possible the system is separable, composed of two copies of the maximally mixed state, or it is maximally entangled, composed of a single EPR pair, or anything in between.
Thus no function of local expectation values can distinguish separable from entangled states in general.
However, purity (which is a function of single site correlation functions), can indeed be used as a bound on the entanglement of a system, again due to monogomy of entanglement. If the local system is not maximally mixed, then it is not maximally entangled, and hence maximum amount of entanglement possible for a system is a monotonic function of its (im)purity.