The whole problem boils down to one thing. Like all traditional grammarians and all laypeople, you have to stop trying to base syntactic decisions on vague intuitions about what modifies what in a common-sense way, like whether something is "giving information about" something specific or the situation as a whole.
Compare this pair:
[1a] Any driver, realizing the danger, would have stopped.
[2a] Any driver realizing the danger would have stopped.
The point is not whether in some vague sense "realizing the danger" gives information about "the driver" by attributing to him the property of realizing the danger.
So why do we call the phrase "realizing the danger" a supplement in
[1a] but a modifier in [2a]?
One reason is that [1a] entails that any driver would have stopped and any driver would have realized the danger. [2a] doesn't: it only says, in effect, that if a driver realized the danger they would have stopped.
In [2a] we have a phrase tightly integrated into the NP, an indispensable part of it. In [1a] we don't: the words "realizing the danger" are a loosely attached interruption of the sentence making a separate and omissible extra entailment.
Loose adjuncts of this kind, the ones we call supplements, can be positioned in various places in the clause:
[1b] Realizing the danger, any driver would have stopped.
[1c] Any driver would, realizing the danger, have stopped.
[1d] Any driver would have stopped, realizing the danger.
But things are totally different with [2a]:
[2b] * Realizing the danger any driver would have stopped.
[2c] * Any driver would realizing the danger have stopped.
[2d] # Any driver would have stopped realizing the danger.
By the asterisks on [2b] and [2c] I mean to indicate that in written form these are actually ungrammatical strings. By the crosshatch before [2d] I mean that although it says something, it would be a bizarre thing to say in a context where [2a] was appropriate; it has totally different truth conditions.
From such considerations, we can conclude that supplements don't always have NP anchors. In [1b], the clause as a whole should probably be considered the anchor.
THIS COULD BE WRONG: grammar is an empirical matter, and there might be a better analysis that treats supplements as modifiers with a special feature requiring extra intonational phrase boundaries (hence the commas).
That is a BIG question, and relates to simplicity and elegance of syntactic
constraints on the form of sentences. Despite the temptation of the mnemonically convenient term "modifier", you can't address that question by exercising your intuition about which phrases modify nouns the way adjectives are supposed to and which phrases modify verbs or clauses the way adverbs are supposed to.