I may have misunderstood the earlier comments (by Rchivers). The type of approach to coordination described with the diagram in the question is indeed how I prefer to view coordination; the diagram indicates what I construe to be a 'small conjunct' approach. The coordinated strings are parallel and the coordination helps mark which strings are interpreted as parallel. Crucially, however, the coordinated strings in this case are NOT constituents, which is a problem for most theories of syntax.
The large conjunct approach, which stands in opposition to such a small conjunct approach, attempts to address such data in terms full underlying clauses. For instance:
(1) [Frank saw {a girl}] and [Bill greeted a girl].
The pointy brackets { } indicate material that has been elided or deleted and is therefore not actually present on the surface. It should be apparent why such an analysis can be called a 'large conjunct' approach. Exactly what the underlying mechanism is called that elides or deletes missing material varies. Some might call it Conjunction Reduction or Right Node Raising or...
The problem with the large conjunct approach is that it suggests a meaning should be present that is often not available. Consider the contrast in meaning across the next two sentences:
(2) [Larry scrounged together] and [Sam borrowed] a total of 1000 dollars.
(3) [Larry scrounged together {a total of 1000 dollars}] and [Sam borrowed a total of 1000 dollars].
The analysis indicated in (3) assumes large conjuncts, that is, the conjuncts are complete clauses underlyingly and an ellipsis or deletion mechanism reduces the first clause down to its surface appearance. In contrast, the analysis indicated in (2) assumes small conjuncts, which means no appeal to ellipsis or deletion is made. It should be apparent that only the small conjunct approach captures the intended meaning, which is such that there is 1000 dollars altogether, not 2000 dollars.
Given that the small conjunct approach comes out ahead in this area, it is nevertheless challenged in major ways. Above all, it has to account for the nature of the non-constituent strings that can and cannot be coordinated.
Finally, note that most phrase structure grammars in general are challenged in major ways by such data, since the implication is that the phrase/constituent is not the crucial unit of analysis that one needs to establish a tractable approach to coordination.