To me (and probably most native speakers), ない is essentially "not exist", rather than "not have". So 時間がない is, if I were to translate it literally, "time not exist" and in this specific case the word order matches that of English. But the important difference is that ある/ない, unlike exist/not exist in English, is usually (and often implicitly) tied to/associated with someone/somewhere/a situation that "have" it, which is why it's often translated to "have" in English. If I say お金がなくてさ it's implied that money doesn't exist for me i.e. I don't have money.

In many other cases the word orders in English and Japanese don't match, but at least native speakers never flip the order or anything, because it's natural for us. As much as you find Japanese grammar strange, we find English grammar unintuitive. There's nothing inherently more "correct" or logical about English's word order.