When I was at primary school, I was given exercises to practice and test my punctuation. Often there were 'trick' sentences to make us think a bit. The one I remember was this:-
time flies we cannot so unsteady is their flight
The order of words is part of the problem: it is poetic or apophthegmatic. It might have meant that 'time flies we cannot ... their ...' ah, that doesn't work. So it must mean that 'timing' flies we cannot their <ie the flies'> flight is so unsteady'. So I'll not put a comma after flies. I'll just put one after cannot.
We need some definitions. The online Cambridge English Dictionary has this.
the symbol ',' used in writing to separate parts of a sentence showing a slight pause, or to separate the single things in a list. But it then goes on to quote a more detailed article in which it describes the use of commas to separate clauses. It provides a useful survey, taken from English Grammar Today and fairly expresses the ins and outs of standard usage.
But we should be clear that punctuation is not exactly part of the written grammar, just as vocal inflection is not not part of the spoken grammar, even though punctuation appears in grammar books. So the people who read the Histories of the Ionian historian, Herodotus, or who learned their parts from a copy one of Aristophanes' comedies were not reading something that suffered from the absence of the punctuation which was only developed several centuries later. Punctuation was not invented until nearly two centuries later by another Aristophanes, chief librarian at the enormous library of Alexandria in Egypt, frustrated by the time it took to operate on the thousands of scrolls in this giant collection. The task involved, among others, coping with the fact that different copies key works differed. This was partly because in mass producing a book, you had slaves around a table taking dictation! Also, theatre directors frequently 'improved' masterpieces of Aeschylus and Sophocles by adding their own ideas. Part of the librarians role was establish an authoritative text. Rapid reading was therefore vital. And Aristophanes suggested to his colleagues that they should annotate the texts to separate the words (that's right, they had not separated words either). They used dots at the end of words, some a the bottom, some in the middle and some at the top. You can read the detail here:
A fuller account of punctuation is to be found in Chapter 11 of the Oxford English Grammar (Sidney Greenbaum). It is well worth a read. He warns against relying too heavily in intonation for the placing of commas.
In relation to the comma, in Chapter 11.3, he says that although punctuation marks are often associated with pauses or the raising or lowering of vocal pitch,
such pauses and also intonation breaks (which mark the ends of intonation units but m)ay sound like pauses to the untrained ear) vary considerably in their placement, depending on factors apart from familiarity with the topic, such as the personality of the speaker, the speaker's mood, the tempo of the speech, and the relationship between the participants in the discourse.
He goes on to give an example of cases where "pauses or intonation breaks occur in speech but punctuation marks are not allowed in writing. As an example he gives:
The question that does remain to be discussed concerns notions of political responsibility.
My own inner voice rises in pitch at the end of "concerns". That all being said, It is also true, at least in British primary (US elementary) schools the children are trained to sing the punctuation like little canaries and will no doubt grow up to transpose what they have been trained to hear in their heads back onto the page when they write.
So despite my scepticism, and given that rigid rules are very difficult to established for something designed mainly for convenience and ease of communication, Lawler is offering a reasonable rule of thumb faute de mieux. But have a good look at the citation in the Cambridge English Dictionary.