Yes you can express any set as a tree (in fact as many possible trees) but if different possible tree representations of that set are equally as valid - then the tree structure isn't really there, it's just an arbitrary interpretation imposed on the data.
If a sentence A B C can be reordered to any permutation, and some parts can be dropped without changing the meaning - then what lets you decide that the proper representation is this tree:
.
/ \
A .
/ \
B C
.
/ \
. C
/ \
A B
Another hint that these kinds of interpretations are wrong is that most succesfull NLP software typically uses word vectors which ignore the tree-like structure and just treats the data as unordered set :)
To understand a sentence it is more important to know all the words in it, than to know how they are nested. Even for analytical languages like English :)
Chomsky isn't concerned with just grammar of written or spoken languages―instead he shows that these grammars map to a universal mental grammar. Different physical languages deal in different ways with constructs of the mental grammar: English expresses the same concepts and relations as inflections, but uses helper words for that. Correspondingly, where you build a grammar tree from words in English, you build it from roots and affixes in inflectional languages. There are languages in which a single word with a bunch of affixes equates to an English sentence.
Regarding “expressing subject (or agent) with verb conjugation”: this should just translate straightforwardly to an implied subject. Even incomplete sentences in whatever language have their implied subjects, verbs and objects―usually deduced from previous speech. Pinker in ‘The Language Instinct’ has a great example of how casual speech is very compact compared to legalese writing where you have to explicitly write out everything in anticipation of ambiguities and adversarial reading.
Chomsky also has a concept of ‘trace’ which is an ‘invisible’ word in a sentence and refers to a previously mentioned word. E.g. in “the spoon that I'm eating soup with,” there's a hidden member: “the spoon that I'm eating soup with <trace>,” and the trace refers to the spoon, helping to build the grammatical tree. I think this concept is hinted at in the article with Gaelic “caught boy caught fish.” Afaiu the ‘trace’ is different from sub-word and implied entities, but it helps to illustrate how the mental grammar isn't the same as written one.