David, I must admit that I'm kind of hopeless with tree diagrams. Could you kindly recommend a good (and simple) reference that would be helpful to me?
Hi, Gilbert—Although I didn't learn how to do syntactic tree diagrams from a texbook, there are some good ones out there (not necessarily simple) by two guys with the first name of Andrew: Andrew Radford and Andrew Carnie.
The parsing I wrongly described as superficial should then be regarded as the correct one:
I didn't mean to be too dismissive of the idea that there is ellipsis there. One old-fashioned theory in generative grammar concerns Conjunction Reduction. I believe your and Gilbert's parsings are in keeping with it.
Logically, the sentence "X has Y and Z," implies, if it is true, that the separate propositions "X has Y" and "X has Z" are true, and, if both are true, their conjunction as independent clauses will be true, too: "X has Y, and X has Z."
But logic and grammar are distinct. In order to express what is logically entailed by a given sentence, we often need to expound at some length, using a variety of expressions. Naturally, it won't work to say that all those words are elided. 
Yesterday, I tried to think of an argument for why the ellipsis-free parsing is needed, independently of the fact that no one would think there are words missing there ("X has Y and Z" can be said unproblematically in a vacuum).
No argument came to me yesterday, but this morning one has. Some kinds of sentences require, for their proper interpretation, that the coordinate structure itself be parsed as the direct object. Consider these examples:
(4) I don't have a sandwich AND an apple.
(5) I don't have a sandwich or an apple.
Sentence (4) means that it is not the case that I have both a sandwich and an apple. It may be true that I have one or the other. Thus, it is not semantically or logically equivalent to "I don't have a sandwich, and I don't have an apple."
Sentence (5) is semantically equivalent to "I don't have a sandwich, and I don't have an apple"; however, the conjunction used is "or." The sentence does not mean the riddle "(Either) I don't have a sandwich, or I don't have an apple."