I didn’t come to offer help.
As far as I can tell, this is how I would analyze this sentence from a grammatical perspective.
I = pronoun
didn’t = aux. verb with “not” for negation.
come = zero infintive verb.
to offer = infinitive acting as an adverb modifying the verb come.
help = ?
I suppose we could consider to offer help to be an adverbial noun phrase, but I’m wondering what the word help would be considered on its own? It certainly can’t be an object since an infinitive acts as a noun so having an object seems illogical.
Here help is the direct object of the infinitive, to offer. Infinitives are always verbs; they are never adverbs.
If you went to see your sister, your sister would be the direct object of that infinitive. Similarly in the case of you wanting to give your sister a call: there she is the indirect object of that infinitive clause.
Nouns don’t take direct object arguments. These do; hence they are verbs.
Infinitives are always verbs. They are non-finite verbs capable of doing anything else a verb can do, particularly in that they take adverbs as modifiers and they take noun phrases as arguments.
To imagine otherwise is going to lead you into contradiction and confusion. Notice how there the infinitive took the adverb otherwise. Nouns don’t take adverbs; verbs do. Just because to imagine otherwise is here the subject of the verb is does not make imagine a noun. That’s just silly. It’s still a verb. The entire clause is a noun phrase, and hence can be the subject. But there are no nouns involved. It’s only just a verb.
This is the same facile error that people so frequently make with gerunds, which are another kind of verb that can be used substantively. That does not make such verbs nouns, and only the entire clause counts as a noun phrase. There is no noun.
There are many kinds of noun phrases, and not all of them even contain any nouns. A noun phrase is a syntactic constituent that can, amongst other things, fulfill the grammatical roles of the subject or object of a verb. Phrases do not have parts of speech.