I disagree with your interviewer, and agree with @JulioJesus's comment. Indeed, since logistic regression is a simple neural network, the interviewer's statement cannot be globally true.
But even in more distinct situations, e.g. a decision tree as the final classifier, I don't see any reason that you shouldn't include both pre-trained embedding features and additional features. Your tree might first select on some of the embedding features, and for each resulting subspace of the embedding proceed with different additional features.
Now it is true that the embedding was generated with the purpose of some classification by the original neural network, so there may or may not be a linear relationship. In that sense, perhaps the embedding features won't work optimally in the classical (esp. a linear) model. But that's a far cry from "cannot use" and "must only use with neural networks," in my opinion.