Aren’t you mixing up “having a model” in the sense of knowing and thinking about one, and having one in the sense of one existing?
You would have to tell me what you mean by “existing.” There is a real (though often informal) distinction between constructive and non-constructive proofs, and this distinction exists in model theory also.
You would have to tell me what you mean by “existing.” There is a real (though often informal) distinction between constructive and non-constructive proofs, and this distinction exists in model theory also.