Confused on my interpretation when negative 'not' + but

[the] purpose of but is semantically to contrast

I think this is your own assumption (perhaps based on some loose explanation of but in dictionaries, etc.), and that this assumption is the cause of your confusion.

I think the contrast conveyed by but in this particular construction is not a semantic one but the one between positive and negative.

The importance of this bill is not in its specific provisions, but in its broader impact.

Which means:

[The importance of this bill is not in its specific provisions], but [it's in its broader impact].

The clause before but is negative and the one after is positive. Hence the contrast.

A different order can be employed:

The importance of this bill is in its broader impact, but not in its specific provisions.

Which means:

[The importance of this bill is in its broader impact], but [it's not in its specific provisions].

Now, the clause before but is positive and the one after is negative. Hence the contrast.

If you were to convey both clauses as positive or negative, you couldn't use but:

??The importance of this bill is in its provisions, but in its impact.

??The importance of this bill is not in its provisions, but not in its impact.

I've removed specific and broader to try to make it less awkward as much as possible, but it's still awkward.

The use of but is to indicate negation of the phrase following. It could be replaced with rather.

The importance of this bill is not in its specific provisions, rather in its broader impact.

From Merriam-Webster Definition of rather than

Used with the infinitive form of a verb to indicate negation as a contrary choice or wish.

Rather than continue the argument, he walked away. He chose to sing rather than play violin

Even though initial clause tells about provisions and second does on impact, purpose of but is semantically to contrast. The two phrases talk about the specifics in contrast to the broader picture. They are well contrasted.

The importance of this bill is not in its specific provisions, but in its broader impact.

OED:

But (conjunction)

III. In a compound sentence, connecting the two coordinate clauses; or introducing an independent sentence connected in sense, though not in form, with the preceding sentence.

11 b.(a) Introducing a statement which is not contrary to, but is not fully consonant with, or is contrasted with, the preceding one (which may be affirmative or negative): nevertheless, yet, however.

2017 Salem (Mass.) News (Nexis) 4 Apr. Cedar shingles are more expensive, but they are also durable.

The importance of this bill is not in its specific provisions, however [it (= the importance of this bill) is] in its broader impact.