Not quite. Nonclassical correlations are strictly stronger than entanglement. The hierarchy is:But as is easy to see, the separable states (i.e., the product states) are quite special, and indeed interactions between the two particles lead to linear combinations which cannot be written as product states, and then you call the parts of the system "entangled". That's the origin of the violation of the Bell inequalities and thus the stronger-than-classical correlations of far-distant parts on an entangled quantum system.
Coherence -> Discord -> Entanglement -> Steering -> Nonclassical Correlations
Local hidden variable models can replicate entanglement and even steering. It's only the final level of the hierarchy they cannot replicate.