Coming from a general linguistics background, everything here seems so confused. An adverb doesn't have to express time or location, it isn't defined by its semantics at all. I'm not an expert in syntax, but I would say an adverb phrase is a modifier of a non-NP, headed by an adverb.

Regardless, as Kris points out, it still isn't the predicative expression here, because it's a modifier, and the predicative expression is a complement. Modifiers can always be left out, while complements never can. So, since "We are a family" is grammatical, very much must be a modifier here, modifying the VP.

Coming from a general linguistics background, the reasoning here seems somewhat confused. An adverb doesn't have to express time or location, it isn't defined by its semantics at all. I'm not an expert in syntax, but one approach might be to say an adverb phrase is a modifier of a non-NP, headed by an adverb.

Regardless, as Kris points out, the AdvP still isn't the predicative expression here. The core reason seems to be that 'very much' isn't doing what we would expect a predicative expression to do, which is to be the thing connected to the subject by the copula.

One explanation may be because it's a modifier, and the predicative expression is a complement. According to some theories of syntax modifiers can always be left out, while complements never can. So, since "We are a family" is grammatical, very much must be a modifier here, modifying the VP.