Mammals already have productivity in motor planning, so given the cognitive architecture to produce that, productivity in language is simply a matter of reuse, not evolution de novo.

Furthermore, human language productivity is severely limited by cognitive constraints; it isn't general-purpose. For example, few people grok "The girl the boy the dog bit hit cried". Instead, we grok language that is more like a complex sequence of goals with refinements, which makes the capability even more akin to postural goals in movement.

Even were it not parallel to existing capacities in other modalities, your arguments would be unconvincing that it could not evolve (only that some arguments about how it did are suspect).