I’m home

It functions as an adverb meaning 'at, to, or in the direction of home'. This also occurs in sentences like "I am going home".

At and to are not adverbs but prepositions.

It's a noun that looks like an adverb because you've dropped out 'now at' or 'here now at' from the sentence.

Um, no. The prepositional phrases 'at home', 'to home' and 'in the direction of home' are adjuncts, or things that modify the verb.

Home in 'I am home', 'Go home' or any equivalent construction is an adverb; like many other words, it does more than one job.

As I frequently say, this is simply the case of a dropped preposition. "At" is implied, thus "home" is, indeed, a noun.

There's no dispute that home is a noun, but it can still be an adverb. Statistically, words in English that act as only one part of speech are a rarity.

That is true. You can have entire phrases or even clauses function as an adverb (or so I was taught), as long as it modifies the verb and pertains to "how," "when," "where," and "to what extent." The following sentences with adverbial phrases and clauses are a few examples:

How: "Johnny answered his mother _in a nonchalant way._"
When: "Sue didn't turn around _when I tapped her shoulder._"

"I am _[at] home._" definitely contains "home" as a noun, an object of the preposition, but also an adverb, since "at home" answers one of the adverb-identifying questions--"where?"

Great. I've been confused myself over this topic, and here I'm facing two different answers just again. But yeah, it helps, as long as we get the meaning I think it's fine either way :D

“'I am home.' Does 'home' function here as a noun or an adverb?"

Does it matter?