Earlier quoted context omitted.
That is already a thing. Simply put it in brackets and insert it into the quote. Overwriting the quote makes the expression unclear, which does require readers seek original expressions or otherwise speculate.
I don’t think that “Heather spoke to [Sam] to the conference” vs “Heather spoke to her [Sam] at the conference” provides all that much meaningful clarity.
In your second example, the quote is clear, verbatim as quotes are intended to be.
In the first example, what has been clear, precise syntax is not a matter of ambiguity.
In your example, it's somewhat clear, but inference is required to understand what was said.
The example given in the article actually does bring genuine ambiguity to what would otherwise be a precise, easily understood expression:
They may groan when Corky, denied the massive funding he’s asked for, tells the town council he’s going to “go home and bite [his] pillow,” the way I do now.
Did Corky say, "go home and bite pillow" broken English style?
Or, did Corky say something else?
Until this construct, we would know because "[his]" would be added into quote context. It would not be a modifier, essentially hiding or changing the quote.
Notably, what Corky did say isn't actually present in the expression, meaning we have to now trust the author interpreting things for us.
Breaking the line with how brackets are used is going to lead to significantly increased ambiguity, and that's going to happen because it will be employed to spin or otherwise manage far less trivial examples, and sorting it out will all take considerably more effort than it all would otherwise be, given brackets are never normally used to modify a quote itself.
And all for a bit of flow? I don't buy it. The whole affair diminishes how robust brackets are and introduced ambiguity into expression that does not add any real value.