The question is concerned with the nature of the strings that can be coordinated. Certainly, coordination (conjunction or disjunction) patterns in many languages similar to how it patterns in English. There is flexibility concerning the strings that can be coordinated, although certain instances of coordination do seem to flow more than others. This is the case with the examples produced in the question.
Verb+object conjuncts occur frequently, e.g.
(1) Frank [killed the chicken] and [ate the frog].
(2) Susan [likes you] but [hates me].
(3) Jim [bought the beer] but [didn’t drink it].
While it is true that subject+verb conjuncts occur much less often, they do occur and can be quite acceptable. Such instances of coordination are known as right node raising (RNR). The next examples are taken from the Wikipedia article on RNR (https://en.wikipedia.org/wiki/Right_node_raising):
(4) [Fred prepares], and [Susan eats], the food.
(5) [Larry has promised], but [Jim refuses], to support reform.
(6) [Jim can], but [Jerry cannot], make the meeting.
(7) [When did he], and [why did he], suffer a setback.
(8) [Sometimes she carefully reads], but [at other times she merely skims], the reports they produce.
Notice the commas (which I have inserted). The presence of RNR is associated with a unique intonation contour, the short pauses aiding processing.
When we look to other languages, we also find great flexibility concerning the strings that can be coordinated. The next examples illustrate some of this flexibility in German (and they would probably not be viewed as instances of RNR):
Certainly, many more similar but distinct such examples can be produced. Consider also the nature of the coordinate structures in the English translations of these examples. The first three involve gapping (Wikipedia: https://en.wikipedia.org/wiki/Gapping), another unique type of coordinate structure, one involving ellipsis.
What we can learn from all these examples is that the coordination mechanism is indeed very flexible. Any claims about the nature of syntactic structures based on coordination should take into account the wide range of phenomena associated with coordination, such as the cases above. My personal view is that coordination as it is usually employed is not a valid diagnostic for identifying the constituents of phrase structure grammars.
Finally, I think the comment at the end of the question is on the right track. I believe that a solid understanding of coordination is promoted by the insight that sentences are produced and processed in the brain (“ordered in our brains”), in an online fashion moving from earlier to later (i.e. left to right in English and other languages that written from left to right).