Can the adverb "always" modify an adjective, or does it always modify a verb?

I agree with your analysis. Although always is classified as an adverb it may play the role of an adjective by modifying the implicit is or are.

In a sentence such as "Always an irritation, noise can be lethal" there is an implied verb, something like "Always {being} an irritation, noise can be lethal".

As to its preceding an adjective, google ngram gives many examples. Here are only three on a vertical scale of about 0 to 0.000005%:

Google ngram

I think you are right in your analysis. In your given sentence, always modifies the verb are.

To support my answer, I present the following logic:

If I take the sentence, He is very honest (Note that the syntax of this sentence is essentially similar to the given sentence's syntax), it can be readily seen that the adverb very modifies the adjective honest. I cannot easily shift about the adverb because it must come near the adjective it modifies. On the other hand, in the sentence He is always honest, I have a grammatical carte blanche of sorts (which is wanting in the former case) to write the sentence either as He always is honest OR more generally as He is always honest, indicating that it modifies the verb and not the adjective.

(A) Yes, you are right.

You're correct in saying that "always" modifies the verb phrase not the adjective in your example, and your reasoning is correct as well.

We can sketch the modificational structure with a standard constituency tree as follows:

(1) a. The green vegetables [T' are [VP always [VP [AP disgusting ]]]]
b. *The green vegetables [T' are [VP [AP always disgusting ]]]

(B) Additional arguments

Here are some additional arguments for this view.
With a non-finite verb, the most natural place for the adverb is between the modal and the main verb, not before the adjective. Likewise, with a lexical main verb, the adverb comes before the verb, not before the adjective.

(2) a. The green vegetables [T' will [VP always [VP be [AP disgusting ]]]]
b. *The green vegetables [T' will [VP be [AP always disgusting ]]]
c. The green vegetables [T' [VP always [VP taste [AP disgusting ]]]]
d. *The green vegetables [T' [VP taste [AP always disgusting ]]]

You cannot easily identify "always disgusting" as a constituent with the question fragment test.

(3) What's your opinion on green vegetables?
*? [AP Always disgusting].

Nor with a gapping test.

(4) *?Kenny found fruit [AP delicious] and green vegetables [AP always disgusting].

You cannot use "always disgusting" in attributive position.

(5) a. His looks are always impeccable.
b. His [AP always impeccable] looks upset me.
c. Green vegetables are always disgusting.
d. *?The [AP always disgusting] vegetables upset me.

(C) "Always" can sometimes modify adjectives

While it is true that "always disgusting" in the question does not form a constituent, this is not necessarily true for all adjectives. See the comment by Peter Shor. See also (5)a. b. above. Here are a few more examples:

(6) a. the [AP always gullible] public
b. the [AP always funny] comedienne
c. my [AP always hectic] schedule
d. an [AP always up-to-date] copy