Learn English – Adverbs in noun phrase coordination

The NP answer is correct. Your objections to it are no good. The Pandora's box argument doesn't make sense -- just because an adverb immediately precedes a noun and there is nothing else in the NP, this doesn't mean the adverb modifies the noun. That is what your argument assumes, and it is just not so. In such cases, the adverb modifies only the NP and not the noun.

Your other objection is that in many cases a given adverb cannot modify a NP. But so what? Adverbs can modify many things, and there are all sorts of restrictions and complexities. Just because adverbs can sometimes modify NPs, it doesn't mean that every adverb can modify every NP.

There is a refinement to the proposal that adjectives are modifiers of nouns that you might like to know about. In McCawley's TSPE (which gives a very interesting taxonomy of adverbs), it is not nouns that are modified by adjectives, but rather N' (N-bar). An N' is (a) a noun alone, (b) a noun with a noun complement (e.g. the P' in "the father of the bride"), or (c) the result of modifying an N' with an adjective, relative clause, or whatever. N' can be replaced by the pro-form "one", as in "The unbroken red ball hit the window before the broken one (one = red ball) did."

One generally assumes that only constituents can be replaced by an indefinite pro-form and that only a constituent can be the antecedent for such a replacement. So since we can go from

I want a big box of apples, but my sister wants only a small box of apples.

to

I want a big box of apples, but my sister wants only a small one.

then "box of apples" must be a constituent. This rules out the structure: [a (Adj) box] [of apples]. It must be instead: a (Adj) [box of apples].