Does every piece of punctuation add syntactic information? [closed]

Does every piece of punctuation add syntactic information?

I am on an endless mission to find a fool-proof usage for commas (for some reason I can't breathe), and in lieu of someone sending me an agreed list of usages...

• My father ate the muffin, the greedy man.

I want to say yes, the comma adds syntactic information above, because it seems like a useful comma (I can't see the noun phrase 'the greedy man' being a complete sentence, yet I feel it does need to be set off as additional and inessential information). But does it? Without the comma it reads, to me, no differently just awkward.

There are various reasons other than to distinguish types of grammatical constructions considered by some to license the use of the comma (I'll limit the discussion to the comma; there are various specialised usages for other punctuation marks, such as factorial (!) and ungrammatical (*), but this would soon become too broad. I'll leave out similar uses of the comma {eg in decimals, large integers ....})

Chicago Manual of Style (13th ed) says in section 5.56:

For ease of reading, it is sometimes desirable to separate two identical or closely similar words with a comma, even though the sense or grammatical construction does not require such separation.

[bolding mine]

• That's what makes Rory, Rory.
• Making the unmissable, unmissable.

And a style guide, One Woman Shop even goes as far as

• Commas: Use commas to create a pause if your breath units are getting too long.

Obviously these recommendations, while unarguably sensible, must take second place to the requirements for clarity, disambiguation, generally understood conventions where a conflict would ensue.